Dr. Trefor Bazett · YouTube · 5:38
This 150-second introductory calculus lecture explains how to think about limits of multivariable functions by first reviewing one-variable limit behavior. The main example is =xy/(), which is undefined at (0,0) because the denominator vanishes there. The presenter warns that a computer-generated surface plot near such a singular point is only an interpolation and should not be treated as exact evidence. Two one-dimensional examples then clarify the logic: a piecewise jump function has no limit at because the left and right limits differ, while a constant function with a hole at still has limit 1 because both sides agree. Returning to the multivariable problem, the video introduces path restriction as a method, chooses the line , substitutes it into the function, simplifies to , and computes the one-variable limit along that path as . The excerpt stops after this single-path calculation and does not state the final conclusion for the full two-variable limit. This 150-second lecture segment analyzes the multivariable function near by restricting it to two straight-line paths. Along the displayed computation gives ; along it gives . A 3D surface plot visually contrasts the two diagonals through the origin. From these unequal path-limits, the lecturer concludes that the full limit at does not exist and names the general criterion the two-path test for nonexistence. The segment ends by raising two unanswered questions: how to define existence of a multivariable limit, and whether agreement along every straight line is sufficient. This 38-second clip contains a short multivariable-calculus reflection followed by an outro. From 0 to 18 seconds, the lecturer displays a boxed proposition stating that if has two different limits along two different paths approaching , then does not exist. Beneath it are two open questions: how to define existence of a limit at a point, and whether agreement along every straight line is enough to guarantee the limit exists. The audio emphasizes the contrast between straight-line paths and a possible curvy path, suggesting that straight-line agreement alone is not treated as conclusive. From 18 to 38 seconds, the scene changes to a studio-like room with a monitor reading 'SUBSCRIBE'; the speaker asks viewers to comment, like the video, and watch more videos in a multivariable calculus playlist. No worked example, proof, or final resolution of the open questions appears within this excerpt.
Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.
Generated from the video's visuals and explanation; not verbatim speech.
The clip opens with a chalkboard-style title, “LIMITS OF MULTIVARIABLE FUNCTIONS,” signaling that the topic is limiting behavior for functions of more than one variable.
The first mathematical object shown is the two-variable function , displayed beside a colored 3D surface plot. The lecturer’s goal is not to evaluate directly, but to ask what happens as approaches .
He immediately identifies the obstruction: at the denominator becomes zero, so the function is undefined there. This motivates a separate question about the limit, because limits concern nearby behavior rather than the value at the point itself.
Using the surface plot, he points to the origin as a visually “pinched” region where two parts of the graph seem to meet. He then adds an important caution: the plotted image is only an approximation, since graphing software interpolates between sampled nearby points and never actually evaluates the undefined point .
To build intuition, the lecture temporarily leaves the multivariable setting and reviews a one-variable piecewise function: for and for . The graph shows a jump at .
From the definition, the left-hand limit is , while the right-hand limit is . Because these one-sided limits disagree, the two-sided limit does not exist. This example establishes the rule that matching approach from both sides is required for a limit to exist.
The next one-variable example changes only the nature of the defect at the point. Now for all , so the function is undefined at , but the graph is otherwise a flat line at height 1.
Here both one-sided limits equal 1: . Therefore even though itself is missing. The contrast with the previous example makes clear that “undefined at the point” and “limit does not exist” are not the same statement.
With that groundwork laid, the lecture returns to the original multivariable function and the problematic point . The speaker proposes a concrete strategy: restrict attention to a particular path through the point, thereby turning the multivariable question into a familiar one-variable limit problem.
He chooses the line , shown in red on the surface plot. Substituting into gives for along that path.
The displayed conclusion is . This is the limit along the single path ; within this excerpt, the video stops after computing that pathwise value and does not yet state whether the full two-variable limit exists.
The segment opens on the worked example with the restriction already displayed. The board shows the reduced one-variable expression and its limit as . The lecturer emphasizes that this trick avoids inventing a new definition: once a path is chosen, the problem becomes an ordinary single-variable limit.
He then changes the restriction to . Algebraically, the substitution inserts minus signs, producing , whose limit is . Visually, the highlighted diagonal on the surface plot switches to the opposite line through the origin, and the lecturer points out that the heights along this new path are much lower, matching the negative limiting value.
Comparing the two results, the lecture draws the key conclusion: because one approach to gives and another gives , the full multivariable limit of at does not exist. The undefined central spot on the surface plot is used as the geometric picture of this path dependence.
A boxed theorem then formalizes the reasoning as the two-path test for nonexistence: if has two different limits along two different paths approaching , then does not exist. The lecturer immediately stresses the limitation of this criterion: it proves nonexistence, not existence.
The clip closes by posing two future questions. First, how should one define what it means for a multivariable limit to exist at a point? Second, if approaching along every straight line gives the same value, does the limit necessarily exist? These questions are shown on screen and left unanswered within this segment.
The clip opens on a static lecture frame with a white-bordered theorem box and two numbered questions beneath it. The displayed mathematical statement is: If has two different limits along two different paths , then does not exist. This is a one-way nonexistence test: unequal path-limits are enough to rule out the full multivariable limit.
While that theorem remains on screen, the speaker shifts attention to Question 2: suppose every straight-line approach gives the same value. The spoken line asks whether a curvy path could change the limit, making clear that the set of all paths is broader than the set of straight lines.
The mathematical point of this segment is contrast. The first displayed fact gives a valid way to prove nonexistence by finding two conflicting paths. The second displayed question asks about the opposite-looking situation, where many paths agree. Agreement along selected families of paths is not presented as a substitute for the full definition of limit existence.
At the cut to the outro, the mathematics stops. The speaker moves into channel promotion, asking viewers to leave questions in the comments, like the video, and continue with a larger multivariable calculus playlist. No answer to either open question is supplied inside this excerpt.
The clip centers on and asks about . The function is not defined at because the denominator vanishes there, so the lecture separates the issue of the function value from the issue of the limiting behavior.
The presenter warns that the 3D graph near is only an approximation. Plotting software interpolates between nearby sampled points and never evaluates the undefined point itself, so the visual “pinch” should motivate analysis rather than replace it.
For the piecewise function when and when , the left-hand limit at is 1 and the right-hand limit is 2. Since these do not match, does not exist.
The second one-variable example, for , is undefined at but approaches 1 from both sides. Hence the two-sided limit exists and equals 1, illustrating the difference between a removable hole and a genuine failure of the limit.
To investigate the multivariable problem, the lecturer restricts the input to a chosen path through the target point. Selecting the line turns into a single-variable expression, making it possible to use ordinary one-variable limit techniques.
Substituting gives , and for this simplifies to . Therefore the limit along that particular path is . The excerpt stops here and does not state the final conclusion for the full two-variable limit.
The lecture studies near the origin. The formula is undefined at , and the whole segment investigates whether a limit exists as by comparing different approaches to that point.
The method shown is to impose a relation between and , such as or , and substitute it into the function. This turns the multivariable question into an ordinary one-variable limit problem, which the lecturer describes as reusing the old one-dimensional concept of a limit.
With , the function becomes . For this simplifies to , so the displayed path-limit is .
With , the numerator becomes while the denominator remains . Thus the restricted expression is , and the displayed path-limit is .
Because the same function approaches along one line through the origin and along another, the lecturer concludes that the full multivariable limit at does not exist. The visual contrast between the two diagonals on the surface plot supports this path-dependence argument.
The general rule stated on screen is: if has two different limits along two different paths , then does not exist. The lecturer names this the two-path test for a limit not existing.
A key caution in the clip is that the two-path test is one-sided in logical force: it can certify that a limit does not exist, but it cannot certify that a limit does exist. Matching values along some paths do not settle the existence question.
After establishing nonexistence by counterexample, the lecture asks how to define what it means for a multivariable limit to exist at a point. This is presented as the next conceptual task, not answered here.
The final prompt asks whether agreement along every straight line through the point would guarantee existence of the full multivariable limit. The clip raises this issue explicitly but leaves the answer for later.
The video explicitly states a sufficient criterion for showing that a two-variable limit fails to exist: if approaches different values along two different paths ending at , then the full limit does not exist. This is a method for disproving existence only; the clip does not claim that matching path-limits prove existence.
Question 1 on the board asks how one defines a limit existing at a point. In this clip it is posed as a conceptual prompt rather than answered, and it sets up the later doubt about whether checking only certain paths is enough.
Question 2 asks whether agreement along every straight-line path forces the limit to exist. The speaker immediately contrasts straight lines with a curvy path, indicating that the full path space matters and that straight-line agreement alone is not treated as settled evidence in this excerpt.
The audio emphasizes a key distinction: even if every straight-line path gives the same result, one can still ask what happens along a nonlinear route. This highlights why multivariable limits are subtler than single-variable limits and why path choice matters.
After 18 seconds the video switches to an outro asking for comments, likes, and playlist viewing. No proof, counterexample, or final answer to the two displayed questions appears within this excerpt.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
Top-left formula reads =xy/().
Speaker says “for a multivariable function like f of x y equal to x y over x squared plus y squared.”
Two-variable function used as the main example in the clip.
Defined for (x,y) with ; not defined at (0,0).
Formula uses variables x and y in xy/().
Speaker refers to approaching the point “zero zero” in the x-y plane.
x, y
Independent real variables of the multivariable function.
Real numbers, subject to the denominator restriction for f.
On-screen text shows lim_{(x,y)->(0,0)} = ???.
Speaker asks what the limit is as x,y gets closer and closer to 0,0.
Limit of the two-variable function as the input approaches the origin.
Questioned value; the clip does not state the final answer.
On-screen piecewise definition: if , and if x>=1.
Speaker describes the same one-dimensional piecewise function.
One-variable piecewise example used to explain nonexistence of a limit at .
All real x; discontinuity at .
On-screen text shows lim_{x->1^-} != _{x->1^+} , followed by “so lim_{x->1} DNE”.
Speaker says the left-hand limit is 1, the right-hand limit is 2, and therefore the limit did not exist.
,\
Left-hand and right-hand limits of the one-variable piecewise function at .
Approach to from below and above.
On-screen text reads , x != 1.
Speaker says the function is not defined at one point but equals 1 on both sides.
,\
One-variable constant function with a single missing point at .
All real x except .
On-screen text shows lim_{x->1^-} _{x->1^+} , then so lim_{x->1} .
Speaker concludes the limit exists and equals 1.
Two-sided limit of the punctured constant function at .
Limit statement about approach to , not the value .
Red text says “Restrict to line ”.
Speaker says “let me demand that y is equal to x” and calls it a restriction.
Chosen path through the origin used to reduce the multivariable problem to one variable.
Line in the xy-plane passing through (0,0).
On-screen derivation writes .
Speaker says after substituting , one is left with x squared over x squared plus x squared, which is x squared over 2x squared.
Single-variable expression obtained by restricting the original multivariable function to the path .
Valid for along the chosen path.
On-screen text shows lim_{x->0} .
Speaker states the resulting one-variable limit equals one half.
Limit of the restricted function along the path as x approaches 0.
Pathwise limit only; the clip does not claim this is the full two-variable limit.
The displayed function is .
The lecturer refers to this as the function under study throughout the clip.
A two-variable real-valued function used as the main example for testing multivariable limits.
The displayed formula is defined when , so ; the video explicitly discusses behavior near .
appears in , in the restricted expressions, and in .
The lecturer says the restriction changes equal to or equal to minus .
x
One independent variable of the two-variable function; after restriction it also serves as the single parameter along the chosen line.
Formula =xy/() appears with a 3D surface plot.
Speaker introduces the topic as the limit of a function as it approaches a point for a multivariable function.
At 28s the screen adds lim_{(x,y)->(0,0)} = ???.
The clip frames the main problem as determining the behavior of =xy/() as (x,y) approaches (0,0), even though the function itself is not defined there. The speaker explicitly connects the issue to the denominator becoming zero at that point.
The function is considered near (0,0).
At (0,0) the denominator is zero, so is undefined.
A 3D surface plot remains on screen while the speaker discusses the pinch at (0,0).
Speaker says the graph is only an approximation because the computer interpolates between nearby points but never exactly at (0,0).
The speaker warns that the displayed surface should not be trusted literally at the singular point because plotting software fills in values by interpolation from nearby sampled points rather than evaluating the undefined point itself.
Applies when visualizing functions with a point of undefined value.
The picture is a computational approximation, not a proof of the limiting value.
Piecewise definition for and for x>=1 is shown.
On-screen text gives lim_{x->1^-} != _{x->1^+} , then “so lim_{x->1} DNE”.
A step graph shows a jump at .
To prepare for the multivariable case, the video revisits a one-dimensional piecewise function and uses mismatched left and right limits at to explain why the two-sided limit does not exist.
The comparison point is .
The left-hand and right-hand limits are different.
On-screen text reads , x != 1.
Then lim_{x->1^-} _{x->1^+} , so lim_{x->1} .
A horizontal line at height 1 is shown with a missing point at .
The second one-dimensional example contrasts with the first: although the function is undefined at , the values on both sides approach the same number, so the limit exists and equals 1.
The function is undefined only at .
Approach from both sides yields the same value.
Speaker says one way to grapple with the problem is to restrict to a particular direction, a particular line.
Red text says “Restrict to line ”.
Derivation on screen substitutes into and simplifies to .
Final displayed line gives lim_{x->0} .
The video introduces a standard strategy for investigating a multivariable limit: choose a path through the target point, substitute the path equation into the function, and study the resulting one-variable limit. Here the chosen path is , producing a restricted expression whose limit as x->0 is .
The path must pass through the point being approached, here (0,0).
Substitution reduces the multivariable expression to one variable.
The computed value is a pathwise limit, not automatically the full two-variable limit.
The board shows .
A 3D surface plot of this function is shown with a highlighted path through the origin.
The clip uses the rational two-variable function as the central example. The lecturer studies its behavior near the origin by restricting the function to different paths and comparing the resulting one-variable limits.
The displayed formula is considered away from because the denominator vanishes there.
The clip focuses on behavior as .
The lecturer says he did not need a new definition for the limit in a higher-dimensional context; he put a restriction on and it became the old one-dimensional concept of a limit.
The board substitutes into to obtain a one-variable expression.
The method demonstrated is to replace one variable by an expression in the other variable, thereby reducing the two-variable problem to a one-variable limit problem. In this clip the chosen restrictions are the straight lines and .
The chosen path must pass through the point where the limit is being tested.
The substitution produces a single-variable expression whose limit can be computed by ordinary one-variable methods.
The displayed computation is and .
The lecturer says the result is one half.
When the function is restricted to the line , the numerator becomes and the denominator becomes , so the restricted expression simplifies to for . Therefore the one-variable limit as along this path is .
This is the limit along the specific path .
The simplification is valid for before taking the limit.
The displayed computation is and .
The lecturer says the result is minus one half and contrasts it with plus one half on the previous line.
When the function is restricted to the line , the numerator becomes while the denominator remains . The restricted expression simplifies to for , so the one-variable limit as along this path is .
This is the limit along the specific path .
The simplification is valid for before taking the limit.
The lecturer explicitly calls the criterion "the two path test for a limit not existing."
A boxed theorem statement appears on screen.
The clip names the criterion the two-path test for a limit not existing. Its purpose is to prove nonexistence of a multivariable limit by exhibiting two approaches to the same point that give different limiting values.
It is used to conclude that a multivariable limit does not exist.
It does not establish existence of the limit.
On-screen text reads "Question 1: How do we define a limit existing at a point?"
The lecturer asks what exactly a limit existing in the multivariable context means.
After showing that different paths can produce different limits, the lecturer raises the next conceptual issue: how to define what it means for a multivariable limit to exist at a point. This is presented as a future topic rather than answered in the clip.
This is a forward-looking question introduced at the end of the segment.
No formal definition is given within this clip.
On-screen text reads "Question 2: Suppose that approaching along any STRAIGHT line gives the same value. Does the limit necessarily exist?"
The lecturer begins introducing the second question after noting that he has been focusing on straight lines.
The second forward-looking question asks whether agreement of the limiting value along every straight line through the point is enough to guarantee existence of the full multivariable limit. The clip poses the question but does not resolve it.
This is a forward-looking question introduced at the end of the segment.
The clip does not provide the answer within the available duration.
Speaker says the function is not defined at , because there becomes a zero in the denominator.
The displayed formula has denominator .
For =xy/(), the function is not defined at (0,0) because the denominator equals 0 there.
=xy/().
Evaluate at (x,y)=(0,0).
At the specific point (0,0).
Screen shows lim_{x->1^-} != _{x->1^+} .
Screen then states so lim_{x->1} DNE.
Speaker says the left and right do not match, so the limit did not exist.
If the left-hand limit and right-hand limit at a point are different, then the two-sided limit at that point does not exist.
The one-sided limits at the point exist or are being compared.
They have different values.
Applied to in the displayed example.
Screen shows , x != 1.
Then lim_{x->1^-} _{x->1^+} , so lim_{x->1} .
Speaker says that in this case the limit did exist and was equal to 1.
For with x != 1, the limit as x approaches 1 exists and equals 1 even though is not defined.
for all x != 1.
Approach from both sides.
At the point .
On-screen derivation shows .
Final line shows lim_{x->0} .
Speaker says the restricted expression is x squared over 2x squared and the limit is one half.
The clip does not state whether this pathwise result determines the full multivariable limit.
After restricting =xy/() to the line , the resulting one-variable limit as x approaches 0 is .
Use the path .
Take x -> 0 along that path.
For approach to (0,0) constrained to the line .
The boxed statement reads: "If has two different limits along two different paths then does not exist."
The lecturer paraphrases this as: if one path gives some limit and another path gives a different limit, then for sure the limit does not exist.
If has two different limits along two different paths , then does not exist.
There are two distinct paths approaching .
The limit of along each path exists.
The two path-limits are different.
For a function of two variables and a point , existence of two unequal path-limits implies nonexistence of the full multivariable limit.
The lecturer concludes there is no limit for this function as goes to .
Earlier displayed path computations give along and along .
For , the limit as does not exist.
Along , the restricted limit is .
Along , the restricted limit is .
These two path-limits are different.
At the specific point for this specific function.
The lecturer says, "Now this test only tells us when a limit does not exist."
The subsequent on-screen questions ask what existence means and whether straight-line agreement suffices.
The two-path test only establishes nonexistence of a multivariable limit; it does not establish existence.
The criterion being discussed is the two-path test for nonexistence.
For the test named in the clip.
The boxed theorem-like statement directly asserts that two different path-limits imply nonexistence of the overall limit.
If has two different limits along two different paths , then does not exist.
is a function of two variables.
There are two different paths approaching .
The limits of along those two paths are different.
For a given and point , existence of two distinct paths with unequal limiting values is sufficient to conclude nonexistence of the full limit.
On-screen Question 2 asks whether agreement along any straight line forces the limit to exist.
The speaker asks whether a curvy path could change the limit or whether only straight lines matter.
The clip does not provide the final answer or counterexample within this excerpt.
Suppose that approaching along any straight line gives the same value. Does the limit necessarily exist?
All straight-line approaches to yield the same limiting value.
Universal over straight-line paths; the status of the implication to full limit existence is posed as a question rather than settled here.
Screen displays lim_{x->1^-} != _{x->1^+} .
Screen then displays so lim_{x->1} DNE.
Speaker explains the left side is 1 and the right side is 2, so they do not match.
Start from the given piecewise definition.
Observed on-screen formula and spoken description.
As x approaches 1 from the left, , so the function uses the branch equal to 1.
Direct reading of the piecewise definition.
As x approaches 1 from the right, , so the function uses the branch equal to 2.
Direct reading of the piecewise definition.
The two one-sided limits are different numbers.
Comparison of the previous two results.
Therefore the two-sided limit at does not exist.
Standard criterion: unequal one-sided limits imply nonexistence of the two-sided limit.
The one-variable piecewise example has no limit at because its left and right limits differ.
Screen displays , x != 1.
Screen displays lim_{x->1^-} _{x->1^+} , then so lim_{x->1} .
Speaker says the function is equal to 1 on both sides and the limit exists and equals 1.
Start from the function that equals 1 everywhere except at the missing point .
Observed on-screen formula and spoken explanation.
From the left of 1, the function value is constantly 1, so the left-hand limit is 1.
Constant behavior on the punctured domain.
From the right of 1, the function value is also constantly 1, so the right-hand limit is 1.
Constant behavior on the punctured domain.
Since both one-sided limits agree, the two-sided limit exists and equals 1.
Matching one-sided limits criterion.
Even though is undefined, the limit as x approaches 1 exists and equals 1.
Red text says Restrict to line .
Screen shows .
Screen shows lim_{x->0} .
Speaker says to plug the x value wherever there is a y, leaving x squared over x squared plus x squared, then x squared over 2x squared.
The clip does not continue to compare other paths or state the final conclusion about the full multivariable limit.
Begin with the original multivariable function.
Observed on-screen formula.
Choose the line as a path through the origin.
Explicitly stated restriction in audio and red on-screen text.
Substitute into numerator and denominator.
Algebraic substitution along the chosen path.
Simplify and .
Elementary algebra.
For along the path, the ratio simplifies to , so the pathwise limit is .
Cancellation of the common factor away from , then taking the limit.
Along the specific path , the restricted one-variable limit is .
The board shows and then .
The lecturer explains that imposing the restriction turns the problem into an old one-dimensional limit computation.
Start from the given two-variable function.
Displayed on the board.
Restrict to the line .
Stated by the red label "Restrict to line ".
Substitute into the function and simplify the denominator.
Direct algebraic substitution shown on the board.
Cancel the common factor for and take the one-variable limit.
Displayed computation and spoken conclusion "which is one half."
Along the path , the limiting value is .
The board updates to and then .
The lecturer says the new restriction introduces minus signs and yields minus one half.
Start again from the same two-variable function.
The original formula remains on the board.
Change the restriction to the line .
Stated by the updated red label "Restrict to line ".
Substitute and simplify.
Direct algebraic substitution shown on the board.
Cancel for and evaluate the one-variable limit.
Displayed computation and spoken conclusion "minus one half."
Along the path , the limiting value is .
The lecturer compares the two results and concludes there is no limit at .
The boxed theorem states that two different path-limits imply nonexistence of the multivariable limit.
One path gives limiting value .
Previously derived along .
Another path gives limiting value .
Previously derived along .
The two path-limits are different.
Immediate comparison of the computed values.
Therefore the full multivariable limit at the origin does not exist.
By the two-path test stated on screen.
Because two different approaches to yield different limits, the multivariable limit of at does not exist.
Main displayed function is =xy/().
A 3D surface plot accompanies the formula throughout the opening and closing sections.
Later the screen asks lim_{(x,y)->(0,0)} =??? and then computes the restriction to .
The clip stops after one path computation and does not present the final verdict for the full two-variable limit.
Determine how to analyze the limit of =xy/() as (x,y) approaches (0,0), where the function is undefined.
=xy/().
The point of interest is (0,0).
The denominator vanishes at (0,0).
Understand what the limit means and begin evaluating it by path restriction.
Identify the function under investigation.
Displayed formula and spoken introduction.
Focus on approach to the origin, not the value at the origin.
Speaker explicitly asks what happens as x,y get closer and closer to 0,0.
Choose a particular path through the origin to reduce the problem to one variable.
Speaker introduces path restriction as a way to grapple with the problem.
Compute the resulting one-variable limit along that path.
Shown algebraically on screen and stated aloud.
The clip obtains the pathwise value along , but it does not state the final answer for the full two-variable limit within this excerpt.
Verification would require comparing multiple paths or applying a rigorous definition of the multivariable limit; that comparison is not shown in this clip.
Piecewise function and one-sided limit statements are shown on screen.
Step graph illustrates the jump at .
Use a one-dimensional example to explain when a limit fails to exist.
for .
for x>=1.
Decide whether lim_{x->1} exists.
Left branch gives value 1.
Piecewise definition.
Right branch gives value 2.
Piecewise definition.
Because the one-sided limits differ, the two-sided limit does not exist.
Standard limit criterion.
The limit does not exist at .
The mismatch 1 != 2 directly verifies nonexistence.
Screen shows , x != 1 and the matching one-sided limits.
Horizontal line with a missing point at .
Show that a function can have a limit at a point where it is not defined.
for all x != 1.
Evaluate lim_{x->1} .
Approach from the left stays on the constant branch 1.
Definition of the punctured constant function.
Approach from the right also stays on the constant branch 1.
Definition of the punctured constant function.
Matching one-sided limits imply the two-sided limit exists and equals 1.
Standard limit criterion.
The limit exists and equals 1, despite being undefined.
Equality of the two one-sided limits verifies existence.
The entire clip works with and computes path limits at the origin.
The lecturer uses this example to motivate the two-path test.
Determine whether exists by comparing limits along different paths.
Path 1:
Path 2:
Target point:
Decide whether the full multivariable limit at exists.
Replace by to obtain a one-variable function along the first path.
Demonstrated method in the lecture.
The limit along equals .
Shown algebraically on the board and stated aloud.
Replace by to obtain a one-variable function along the second path.
Demonstrated method in the lecture.
The limit along equals .
Shown algebraically on the board and stated aloud.
The two path-limits disagree.
Direct comparison of the computed values.
Therefore the multivariable limit at the origin does not exist.
By the two-path test for nonexistence.
The limit does not exist at .
Verification is internal to the method: two explicit paths approaching the same point produce different limiting values, which triggers the stated two-path test.
Black chalkboard background with animated handwritten title text appearing in sequence.
Title reads LIMITS OF MULTIVARIABLE FUNCTIONS.
Chalkboard background
Handwritten title text
Words appear sequentially until the full title is visible.
No mathematical formula is shown yet.
No speaker is visible.
This segment serves as a title card introducing the topic before the lecture content begins.
Left side shows a 3D surface plot with axes labeled roughly from -2 to 2 in x and y and about -1 to 1 vertically.
Top-left formula =xy/() is visible.
Pink annotation Problematic at (0,0)! points toward the origin region.
At 28s the line lim_{(x,y)->(0,0)} = ??? appears.
Exact axis tick labels are small; the broad ranges are clear but fine readings are approximate.
3D surface plot
Formula =xy/()
Pink arrow annotation
Limit question text
Presenter on the right
The limit question appears after the initial setup.
The presenter gestures toward the pinch region of the surface.
The surface remains the same example function throughout this interval.
The highlighted problematic location is (0,0).
The visualization supports the verbal point that the origin is a singular pinch point and that the plotted surface is only an interpolation-based approximation near that point.
2D graph shows a horizontal red segment at for and another at for x>=1.
Piecewise definition and one-sided limit statements are displayed above the graph.
Coordinate grid
Two horizontal red segments
Piecewise formula
One-sided limit formulas
The graph replaces the earlier 3D surface.
Text builds from the function definition to the mismatched one-sided limits and DNE conclusion.
The discontinuity is located at .
The left level is 1 and the right level is 2.
The picture concretizes the abstract rule that unequal left and right limits produce nonexistence of the two-sided limit.
A horizontal red line at is shown with a missing point at .
Text reads , x != 1 and then the matching one-sided limits leading to limit = 1.
Coordinate grid
Horizontal red line at
Gap at
Limit formulas
The previous jump graph is replaced by a single-level graph with a puncture.
The text changes from DNE to an existing limit equal to 1.
The function value is 1 everywhere it is defined in the displayed window.
The only exceptional point is .
The visual contrast with the previous example shows that missing the value at a point is different from having incompatible approaches from the two sides.
The 3D surface plot returns on the left.
Red text says Restrict to line .
A red line is drawn across the surface corresponding to the path .
Derivation text appears stepwise: , then lim_{x->0} .
The exact geometric thickness and endpoints of the red path line are visual approximations, but the intended path is explicit.
3D surface plot
Red path line
Restriction label
Algebraic derivation text
Presenter
The scene returns from 1D examples to the original 2D function.
A red line is overlaid to indicate the chosen path.
The algebraic reduction and final pathwise limit appear in sequence.
The underlying function remains =xy/().
The chosen path remains once introduced.
The animation links the abstract substitution to a concrete curve on the surface, showing how a multivariable problem is reduced to a one-variable limit along that path.
A 3D surface plot of is shown with a red line marking the path through the origin.
The red label reads "Restrict to line ".
The exact axis orientation is visible but not fully labeled in every frame; the mathematical meaning is clear from the formulas and narration.
3D surface of
red line on the surface
coordinate axes
formula panel
The red highlighted path corresponds to .
The displayed algebra updates from the general function to the restricted one-variable expression.
The underlying function remains .
The point of interest is the origin.
The visual emphasizes that approaching the origin along the line produces a definite height trend corresponding to the computed limit .
The red highlighted line changes to the opposite diagonal, matching the label "Restrict to line ".
The algebra changes to and the limit to .
same 3D surface
new red line on the opposite diagonal
updated formula panel
The highlighted path changes from to .
The displayed restricted expression changes sign in the numerator.
The computed limit changes from to .
The surface itself does not change.
The target point remains the origin.
The visual contrast shows that a different straight-line approach to the same point can lead to a different limiting height, supporting path dependence.
A boxed statement appears: "If has two different limits along two different paths then does not exist."
The lecturer names this the two-path test for a limit not existing.
boxed theorem text
lecturer speaking beside it
The clip shifts from the worked example to a general criterion.
The example function has already established unequal path-limits.
The box formalizes the reasoning used in the example: unequal path-limits imply nonexistence of the full multivariable limit.
Text appears reading "Question 1: How do we define a limit existing at a point?" and then "Question 2: Suppose that approaching along any STRAIGHT line gives the same value. Does the limit necessarily exist?"
The lecturer introduces these as two future questions.
two on-screen question statements
lecturer
The presentation moves from the nonexistence test to open conceptual questions.
The discussion remains about multivariable limits and path-based reasoning.
The visuals signal that the current clip proves nonexistence by counterexample paths, but does not yet define existence or settle whether straight-line agreement is sufficient.
A lecturer stands before a dark chalkboard-style background with a white-bordered box at upper left containing the two-path theorem text, plus two numbered questions below.
The speaker gestures while talking; no mathematical diagram changes occur during this interval.
Lecturer
White-bordered theorem box
Question 1 text
Question 2 text
Chalkboard-style background
Only the speaker's hand gestures change.
The displayed mathematical text remains fixed throughout this interval.
The theorem statement about two different path limits stays visible.
Question 1 and Question 2 remain on screen unchanged.
The visual layout separates a known sufficient criterion for nonexistence from two unresolved conceptual questions about how to define limit existence and whether straight-line agreement is enough.
At 18 seconds the scene cuts to a different room with acoustic foam panels, a desk microphone, a laptop, and a monitor displaying 'SUBSCRIBE'.
The speaker points downward and later gives a thumbs-up while delivering the outro.
Lecturer in different shirt
Monitor with 'SUBSCRIBE'
Desk microphone
Laptop
Acoustic foam wall panels
Background changes completely from chalkboard-style lecture frame to office/studio setup.
Speaker gestures shift from explanatory hand motions to pointing and thumbs-up.
No new mathematical content appears after the cut.
This visual transition marks the end of the mathematical discussion and the start of channel promotion.
Speaker says when he tries to graph something like this, it is only an approximation because the computer is interpolating between nearby points but never equal to 0,0.
One may think the computer-generated graph shows the true value or limiting behavior at (0,0) because the surface appears continuous there.
The speaker explicitly warns that the plot is interpolated from nearby sampled points and does not evaluate the undefined point itself, so the picture alone should not settle the limit question.
Screen shows , x != 1 and then lim_{x->1} .
Speaker contrasts a function not defined at one point with the existence of its limit there.
Students may conflate “the function is not defined here” with “the limit does not exist here.”
The clip’s second one-variable example shows that even when is missing, the limit can still exist if both sides approach the same value.
The clip computes only one path, , and obtains .
Speaker introduces path restriction as one way to start grappling with the problem, not as a completed proof of the full limit.
This caution is an analyst-added clarification; the excerpt itself does not explicitly warn against overinterpreting a single path.
One might infer from the displayed value that the full limit lim_{(x,y)->(0,0)} has already been determined.
Editorial clarification: the excerpt only evaluates the function along the chosen path . To decide the full two-variable limit, one would need additional argument or comparison with other paths, which is not shown in this clip.
The lecturer explicitly says the test only tells us when a limit does not exist.
The follow-up questions ask what existence means and whether straight-line agreement suffices.
One might think that checking paths and getting matching values proves the multivariable limit exists.
The clip states that the two-path test only proves nonexistence when two paths give different limits. It does not establish existence, and the lecturer raises the separate question of what existence really means.
Question 2 asks whether agreement along any STRAIGHT line guarantees existence.
The lecturer notes that he has been focusing on straight lines and introduces this as a remaining issue.
The clip poses the question but does not answer it within the available duration.
One might assume that if every straight line through the point gives the same limiting value, then the full multivariable limit must exist.
The video explicitly raises this as an open question rather than asserting it as true. Within this clip, straight-line agreement is not shown to be sufficient.
Question 2 explicitly asks whether agreement along any straight line implies the limit exists.
The speaker follows by asking what happens if one approaches along a curvy path and whether that could change the limit.
One might think that if every straight-line approach gives the same value, then the multivariable limit must exist.
The video treats this as an open question and immediately contrasts straight lines with curvy paths, signaling that straight-line agreement alone is not presented as sufficient evidence for the full limit.
The displayed statement only concludes nonexistence from two different path limits.
From the wording of the theorem, the implication runs from unequal path limits to nonexistence, not the reverse.
One might incorrectly read the two-path rule as saying that matching path limits prove the limit exists.
The displayed proposition is one-directional: different path limits imply nonexistence. It does not state that equal path limits imply existence.
Speaker says, “To help me understand this, let’s go back to the one-dimensional case.”
The video switches from the 2D function to two 1D examples before returning to the 2D problem.
The clip uses one-variable limit reasoning as preparation for understanding pathwise restriction in the multivariable example.
First 1D example ends with DNE because one-sided limits differ.
Second 1D example ends with limit = 1 despite the function being undefined at the point.
The two one-variable examples are deliberately contrasted: mismatched sides destroy the limit, while a lone missing point does not.
Speaker returns to the multivariable case and says one way to grapple with it is to restrict to a particular line.
The restriction is applied directly to =xy/().
The path-restriction method is introduced specifically as a tool for investigating the central multivariable limit question.
After substituting , the problem becomes lim_{x->0} .
Speaker says the restriction makes the multivariable function into a single-variable function that we know how to do.
Once a path is chosen, the multivariable question is handled using ordinary one-variable limit techniques, which the preceding examples review.
The lecturer says putting a restriction made it the old one-dimensional concept of a limit.
The board converts into a one-variable expression after substituting .
The method of restricting to a path is applied to compute the one-variable limit along .
The lecturer moves from the two computed path-limits to the named criterion.
The boxed theorem generalizes the example's reasoning.
The worked example supplies the concrete unequal path-limits that motivate and instantiate the general two-path test.
The lecturer concludes there is no limit at after comparing the two path results.
The theorem box states that different path-limits imply nonexistence.
The general two-path test is applied to the specific function to conclude that its limit at the origin does not exist.
After saying the test only proves nonexistence, the lecturer asks what existence means.
Question 1 appears on screen.
The limitation of the test motivates the separate question of how to define existence of a multivariable limit.
The lecturer notes he has been focusing on straight lines and introduces the second question.
Question 2 asks whether agreement along any straight line suffices.
The clip's method uses straight-line restrictions, and the final question contrasts that with the unresolved issue of whether straight lines alone are enough to decide existence.
The boxed theorem concerns two different paths and nonexistence.
Question 2 asks about all straight lines giving the same value and whether the limit then exists.
The first item gives a valid way to disprove existence by finding unequal path limits, while the second item asks about the converse-style situation where many paths agree; the contrast highlights that agreement along selected paths is not treated like the proven nonexistence criterion.
Question 1 asks how to define a limit existing at a point.
Question 2 asks whether straight-line agreement forces existence.
Answering whether straight-line agreement suffices depends on having a precise definition of what it means for a multivariable limit to exist.
The theorem is stated for .
The two-path nonexistence method is applied specifically to the multivariable limit notation shown on screen.
Screen shows =xy/() and later lim_{(x,y)->(0,0)} =???.
Speaker introduces the limit of a multivariable function as it approaches a point.
Speaker says there becomes a zero in the denominator at , .
Denominator is .
Speaker says the graph is only an approximation because the computer interpolates between nearby points.
Screen shows lim_{x->1^-} != _{x->1^+} , so lim_{x->1} DNE.
Screen shows , x != 1 and then lim_{x->1} .
Speaker says one way to grapple with the problem is to restrict to a particular direction, a particular line.
Red text says Restrict to line .
Screen derives and then lim_{x->0} .
Only one path, , is evaluated in the excerpt.
Speaker presents path restriction as a way to start grappling with the problem.
The excerpt does not explicitly discuss the logical insufficiency of a single path for proving existence.
The boxed theorem statement is visible.
The lecturer names the two-path test.
The two displayed path-limits are and .
The lecturer contrasts the previous line with the new line.
The board shows .
The board shows .
Covered · Title card only; no mathematical content beyond topic identification.
Covered · Introduction of =xy/(), the question of its limit at (0,0), and the warning that the plotted surface is interpolated near the singularity.
Covered · One-variable piecewise example showing unequal one-sided limits and nonexistence of the two-sided limit.
Covered · One-variable punctured constant example showing that a limit can exist even when the function is undefined at the point.
Covered · Return to the multivariable example and computation of the restricted limit along the path , yielding .
Covered · Final fraction of a second contains no new mathematical event beyond the already displayed pathwise result.
Covered · Opening computation of the limit along with surface plot and algebra.
Covered · Switch to the path , computation of the second limit, and visual contrast between the two paths.
Covered · General statement of the two-path test, conclusion of nonexistence for the example, and the two closing conceptual questions.
Covered · Mathematical content consists of the displayed two-path nonexistence rule and two open conceptual questions about multivariable limits.
Covered · No mathematical content is introduced in this interval; it is an outro asking for comments, likes, and playlist viewing.
Reviewed subject paths
Continue with related videos in the same subject.
YouTube8:56
YouTube4:33
YouTube8:53