mathvideo•A complete graph exercise comparing left and right limits at the same input and explaining why their disagreement rules out a finite two-sided limit.
Can a function have two different approached values at the same input? This graph exercise reads the left-hand limit at 3 as 4 and the right-hand limit as 1, then uses their disagreement to conclude that the two-sided limit does not exist. Moving points and guide lines separate the curve’s nearby behavior from the filled value f(3)=1. The notes clarify that finite sample readings illustrate a limit rather than prove a general formula, and that open or filled markers alone do not determine limiting behavior.
Generated from the video's visuals and explanation; not verbatim speech.
The video presents a graph of a piecewise function on a Cartesian coordinate plane. The goal is to determine the limit of as approaches 3. To do this systematically, the problem is broken down into evaluating the left-sided limit and the right-sided limit.
The instructor writes the notation for the left-sided limit: . The small minus sign superscript on the 3 is explained as indicating that will approach 3 from values strictly less than 3 (e.g., 1, 2, 2.5, 2.99).
In the illustrated near-point portion of the left branch, start at , where , and move towards 3. The curve approaches the open endpoint . This local reading does not say the entire branch is monotone.
Because the curve approaches a -value of 4 as approaches 3 from the left, the left-sided limit is determined to be 4. This is written as . The solid dot at , which represents the actual function value , is ignored for this specific limit calculation, demonstrating that limits describe behavior near a point, not necessarily at the point itself.
Continue with the same graph and consider the right-hand approach to 3. The three statements on the board separate the two-sided, left-hand and right-hand questions.
Sample the right branch at 5, 4 and 3.5. These nearby illustrated points move toward the target height as the input moves toward 3; the exercise uses the displayed branch behavior rather than a finite-point general proof.
Near input 3, the right branch approaches height 1. The guide lines and final written equality record this right-hand limit. The filled point alone is not the justification.
For a finite two-sided limit in this interval setting, both finite one-sided limits must agree. The board gives the left value as 4 and the right value as 1.
The values 4 and 1 differ, so the left and right approaches have no common limiting value. The board therefore marks the two-sided limit as nonexistent.
If the two finite one-sided limits were the same, their common number would give the two-sided limit. Here the mismatch at input 3 completes the answer.
A left-sided limit is denoted by a minus superscript on the target value, such as . It describes the value that the function approaches as the input gets arbitrarily close to from the left side, meaning from values strictly less than .
To find a one-sided limit from a graph, trace the curve from the appropriate side towards the target x-value. Observe the y-value that the curve approaches. The actual y-value of the function at the target x-coordinate (indicated by a solid dot) does not affect the limit; only the behavior of the curve near that x-value matters.
In this displayed piecewise graph, the open circle at (3,4) marks the left branch endpoint and the solid point at (3,1) gives the function value. The left limit is read from the nearby branch, not from the solid point. Other graphs can have limits at solid points too.
Approach the target through inputs on its right. Here the sampled inputs 5, 4 and 3.5 illustrate the right branch approaching height 1, leading to the displayed right-hand limit at 3.
As the input approaches 3 from below, the left branch approaches height 4. The open endpoint marks the approached position in this graph.
For an ordinary finite two-sided limit on a real interval around the target, both finite one-sided limits must exist and equal the same number. Agreement gives the two-sided limit; disagreement rules it out. This exercise has unequal finite values.
At the target input 3, the approached values are 4 on the left and 1 on the right. Their inequality is the reason the two-sided limit does not exist.
The open point (3,4) and filled point (3,1) mark endpoint inclusion in this graph. Determine each limit by following the corresponding nearby branch, not by the marker type alone.
Editorial reminder: the single value f(3) does not determine a limit. Here the filled point is (3,1), but the left branch approaches 4. The unequal one-sided limits defeat a two-sided limit.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
The expression is written in the prompt at the top right and spoken throughout as the function whose limit is being considered.
The blue curve on the coordinate plane is labeled and represents the graph of the function.
f(x)
The value of the function f at input x.
The shown function is examined near input 3; its full domain is not specified by the viewing window.
The variable x appears in the expressions and .
The horizontal axis of the coordinate plane is labeled x.
x
The independent variable or input value of the function f.
x \in \mathbb{R}
The notation is written on the screen.
The spoken explanation in this interval discusses The limit operator, indicating the value that a function approaches as the input approaches a certain point..
\lim
The limit operator, indicating the value that a function approaches as the input approaches a certain point.
N/A
A small minus sign is written as a superscript to the 3 in .
The spoken explanation in this interval discusses Indicates a left-sided or left-hand limit, meaning the input approaches the target value from values strictly less than the target..
^-
Indicates a left-sided or left-hand limit, meaning the input approaches the target value from values strictly less than the target.
N/A
The function is written as f(x) in the limit expressions on the right side of the screen.
The graph on the left is labeled with an italic f near the curve.
f(x)
The dependent value of the plotted function at input x.
Displayed graphically over roughly x=-9 to x=9; the clip focuses on behavior near x=3.
The variable x appears in x -> 3, x -> 3^-, and x -> 3^+.
The spoken explanation in this interval discusses The independent variable whose values approach 3 from either side..
x
The independent variable whose values approach 3 from either side.
Real-valued horizontal coordinate on the graph.
The top-right expression reads lim_{x -> 3} f(x).
The spoken explanation in this interval discusses The two-sided limit of f(x) as x approaches 3..
lim_{x -> 3} f(x)
The two-sided limit of f(x) as x approaches 3.
An ordinary finite two-sided limit is defined when both finite one-sided limits exist and have the same real value.
The middle-right expression reads lim_{x -> 3^-} f(x) = 4.
The spoken explanation in this interval discusses The left-hand limit of f(x) as x approaches 3 from values less than 3..
lim_{x -> 3^-} f(x)
The left-hand limit of f(x) as x approaches 3 from values less than 3.
One-sided limit from the negative direction relative to 3.
The bottom-right expression is completed during the clip as lim_{x -> 3^+} f(x) = 1.
The spoken explanation in this interval discusses The right-hand limit of f(x) as x approaches 3 from values greater than 3..
lim_{x -> 3^+} f(x)
The right-hand limit of f(x) as x approaches 3 from values greater than 3.
One-sided limit from the positive direction relative to 3.
The superscripts '-' and '+' appear on the 3 in the one-sided limit notation.
The spoken explanation in this interval discusses Direction markers indicating approach to 3 from below (left) or above (right)..
3^- , 3^+
Direction markers indicating approach to 3 from below (left) or above (right).
Used only inside one-sided limit notation.
The expression is written on the screen.
The spoken explanation in this interval discusses Left-sided limit.
A left-sided limit, denoted by a minus superscript on the target value (e.g., ), describes the value that the function approaches as the input gets arbitrarily close to the target value from the left side (i.e., from values strictly less than the target).
The input approaches c through domain values strictly below c.
This displayed graph has a branch defined on an interval immediately to the left; more generally c must be a left accumulation point of the domain.
The spoken explanation in this interval discusses Evaluating limits graphically.
A red dot moves along the blue curve from left to right towards x=3, and dashed lines project its position to the x and y axes.
To evaluate a one-sided limit from a graph, trace the curve starting from an x-value on the appropriate side of the target and move towards the target x-value. Observe the corresponding y-value that the curve approaches. The actual y-value of the function at the target x-coordinate (if it exists) does not affect the limit.
A visual graph of the function is available near the target x-value
The spoken explanation in this interval discusses Right-hand limit read from a graph.
The bottom-right expression is written progressively and ends as lim_{x -> 3^+} f(x) = 1.
The graph shows a solid blue point at (3,1) on the right branch, with dashed guide lines to y=1 and x=3.
To estimate lim_{x -> 3^+} f(x), follow the branch of the graph for x>3 as x moves toward 3 from the right and observe the y-value approached by f(x). In this clip, the sampled points x=5, x=4, and x=3.5 lead the speaker to conclude the approached value is 1.
Use the portion of the graph with x>3.
The conclusion is based on visual estimation from the displayed graph.
The spoken explanation in this interval discusses Criterion for existence of a two-sided limit.
The screen displays lim_{x -> 3^-} f(x) = 4 and lim_{x -> 3^+} f(x) = 1 beneath the two-sided expression lim_{x -> 3} f(x).
The spoken explanation in this interval discusses Criterion for existence of a two-sided limit.
A two-sided limit exists only when the left-hand and right-hand limits are equal. If those one-sided limits approach different numbers, the two-sided limit does not exist.
The function is considered on both sides of the target in a real interval.
Both finite one-sided limits must exist and equal the same real number for a finite two-sided limit.
At x=3 the graph has an open circle at (3,4) on the left branch and a solid circle at (3,1) on the right branch.
The written one-sided limits assign different values: 4 from the left and 1 from the right.
The spoken explanation in this interval discusses Visual signature of a jump discontinuity at x=3.
The graph shows a break at x=3: the left branch approaches height 4 but does not include that endpoint, while the right branch starts at height 1 with a filled endpoint. This mismatch in approached heights is why the two-sided limit fails to exist.
Interpretation is based on the displayed graph near x=3.
Open circle indicates exclusion of that endpoint; solid circle indicates inclusion.
The middle-right handwritten statement reads lim_{x -> 3^-} f(x) = 4.
The left branch approaches an open circle at (3,4), with dashed guides to y=4 and x=3.
\lim_{x \to 3^-} f(x) = 4
Use the branch of the graph with x<3.
Approach x=3 from the left.
For x approaching 3 through values less than 3.
The spoken explanation in this interval discusses Right-hand limit at x=3.
The bottom-right handwritten statement is completed as lim_{x -> 3^+} f(x) = 1.
The right branch has a solid point at (3,1) with dashed guides to y=1 and x=3.
\lim_{x \to 3^+} f(x) = 1
Use the branch of the graph with x>3.
Approach x=3 from the right.
For x approaching 3 through values larger than 3.
The spoken explanation in this interval discusses Nonexistence of the two-sided limit at x=3.
The top-right expression lim_{x -> 3} f(x) is annotated with 'does not exist'.
The displayed one-sided limits are unequal: 4 and 1.
\lim_{x \to 3} f(x) \text{ does not exist.}
\lim_{x \to 3^-} f(x) = 4
\lim_{x \to 3^+} f(x) = 1
4 \neq 1
Concerns the two-sided approach to x=3.
The spoken explanation in this interval discusses Derivation of the left-sided limit from the graph.
The final result is written on the screen.
The red dot traces the curve and the horizontal dashed line approaches y=4.
Identify the target x-value (3) and the direction of approach (from the left, or values less than 3).
Definition of a left-sided limit.
Read the displayed near-point samples as graph estimates, not exact values of an unspecified algebraic function.
Graphical evaluation of function values.
Observe that as x gets closer and closer to 3 from the left, the y-values of the curve approach 4. The open circle at (3, 4) confirms this target y-value.
Visual convergence of the graph to the point (3, 4).
The left-sided limit of f(x) as x approaches 3 is 4.
The spoken explanation in this interval discusses Estimating the right-hand limit from sample points.
The cursor traces along the right branch toward the solid point at (3,1).
The bottom-right limit expression is completed as lim_{x -> 3^+} f(x) = 1.
The exact intermediate y-values at x=5, x=4, and x=3.5 are not written numerically on screen; only their trend is discussed.
Identify the task as finding the limit as x approaches 3 from values larger than 3.
Stated directly in the audio at the beginning of the clip.
Look at the graphed point on the right branch for x=5 and note its height.
The spoken graph explanation supports this step at the corresponding interval.
Move closer to x=3 and inspect the corresponding point on the same branch.
The spoken graph explanation supports this step at the corresponding interval.
Take an even closer sample and observe that the function value is a little under 2.
The spoken graph explanation supports this step at the corresponding interval.
Conclude that as x approaches 3 from the right, f(x) approaches 1.
The spoken graph explanation supports this step at the corresponding interval.
The right-hand limit is estimated visually as 1.
The spoken explanation in this interval discusses Using one-sided limits to decide the two-sided limit.
The screen shows lim_{x -> 3^-} f(x) = 4 and lim_{x -> 3^+} f(x) = 1 under lim_{x -> 3} f(x).
The phrase 'does not exist' is written next to the two-sided limit.
Recall the criterion that a two-sided limit exists only if the left-hand and right-hand limits are equal.
The spoken graph explanation supports this step at the corresponding interval.
Record the left-hand limiting value shown on the board.
Visible in the middle handwritten equation.
Record the right-hand limiting value just derived.
Visible in the bottom handwritten equation and supported by the preceding graph reading.
Compare the two one-sided limits and note they are different.
Immediate numerical comparison of the displayed values.
Apply the existence criterion to conclude nonexistence of the two-sided limit.
The speaker states this conclusion directly after contrasting the left and right approaches.
Because the one-sided limits are unequal, the two-sided limit at x=3 does not exist.
The entire video focuses on evaluating the limit for the specific piecewise function graphed on the coordinate plane.
The prompt and the partial solution are written on the screen.
Given the graph of a function f, determine the value of .
The graph of the function f is provided on a Cartesian coordinate system.
The function has a discontinuity at x=3, with an open circle at (3, 4) and a closed circle at (3, 1).
Find the numerical value of the left-sided limit as x approaches 3.
Start at an x-value less than 3 (e.g., x=2.5) and move along the blue curve towards x=3.
This visually represents x approaching 3 from the left.
In the sampled near-target portion, the displayed values move from about 5 through about 4.5 towards 4; this does not claim monotonicity across the whole branch.
Reading the vertical position of the curve relative to the y-axis.
The curve approaches the open circle located at the coordinates (3, 4). Therefore, the y-value it approaches is 4.
The definition of a limit depends on the behavior of the function near the point, not at the point itself.
4
The visual tracing of the graph clearly shows the curve converging to the y-coordinate of the open circle at x=3, which is 4.
The spoken explanation in this interval discusses Determining whether lim_{x -> 3} f(x) exists from a graph.
A coordinate graph with a jump at x=3 is used throughout.
Three limit statements are written on the right: the two-sided limit, the left-hand limit, and the right-hand limit.
Given the displayed graph of f, determine the right-hand limit at x=3 and decide whether the two-sided limit exists.
Graph of f with a break at x=3.
Left branch approaches an open circle at (3,4).
Right branch approaches a solid circle at (3,1).
Already written on screen: \lim_{x \to 3^-} f(x) = 4.
Find \lim_{x \to 3^+} f(x) and use the one-sided limits to decide the status of \lim_{x \to 3} f(x).
Follow the graph for inputs larger than 3 as they move toward 3.
This is the definition of a right-hand limit.
Sample several points on the right branch to see the trend of f(x).
The speaker explicitly names these x-values while tracing the graph.
Conclude from the trend and the solid endpoint at (3,1) that the right-hand limit is 1.
Supported by both the spoken estimation and the final handwritten equation.
Compare the left-hand and right-hand limits.
Both values are displayed on the board.
Since the one-sided limits differ, the two-sided limit fails to exist.
This is the criterion stated by the speaker and written on screen.
\lim_{x \to 3^+} f(x)=1, and therefore \lim_{x \to 3} f(x) does not exist.
Check that the left-hand and right-hand limiting values are unequal; unequal one-sided limits imply nonexistence of the two-sided limit.
A red dot appears on the blue curve and moves rightward towards x=3. Dashed red lines connect the dot to the x and y axes, updating dynamically.
Red dot
Blue curve
Dashed red lines
x-axis
y-axis
The red dot moves along the blue curve from left to right.
The vertical dashed line moves rightward along the x-axis towards x=3.
The horizontal dashed line moves downward along the y-axis towards y=4.
The red dot remains on the blue curve.
The dashed lines always connect the red dot perpendicularly to the axes.
This animation visually demonstrates the concept of a left-sided limit by showing how the function's output (y-value) changes as the input (x-value) approaches the target from the left.
The left side of the screen shows a Cartesian graph with x-axis from -9 to 9 and y-axis from -9 to 9.
Near x=3 there is an open circle at (3,4) on the left branch and a solid circle at (3,1) on the right branch.
Dashed guide lines connect these endpoint circles to the axes at x=3 and the corresponding y-values 4 and 1.
Coordinate axes
Graph of f
Open circle at (3,4)
Solid circle at (3,1)
Dashed guide lines to axes
The cursor moves along the right branch from larger x-values toward x=3.
The bottom-right limit expression is progressively completed until it reads =1.
The top-right two-sided limit is annotated with 'does not exist'.
The left branch continues to indicate approach to height 4.
The right branch continues to indicate approach to height 1.
The mismatch between these two heights remains visible throughout.
The picture encodes different one-sided limiting behaviors at the same x-value, which is the geometric reason the two-sided limit fails.
The bottom-right expression is built step by step from 'lim' to 'lim_{x -> 3^+} f(x)' and finally to '=1'.
The spoken explanation in this interval discusses Stepwise construction of the right-hand limit statement.
Handwritten limit notation
Cursor/pen position on the right panel
First the limit operator appears.
Then x -> 3^+ is added.
Then f(x) is written.
Finally =1 completes the statement.
The target point of approach remains x=3 from the right throughout.
The animation turns a verbal graph-reading procedure into formal one-sided limit notation.
After comparing the one-sided limits, the words 'does not exist' are written next to lim_{x -> 3} f(x).
The spoken explanation in this interval discusses Annotation of nonexistence on the two-sided limit.
Top-right expression lim_{x -> 3} f(x)
Handwritten phrase 'does not exist'
The previously bare two-sided limit expression receives a final status annotation.
The one-sided limit values 4 and 1 remain unchanged on the board.
The visual annotation records the logical outcome of the comparison between one-sided limits.
The graph shows a closed blue circle at (3, 1) and an open blue circle at (3, 4). The limit being calculated is 4.
This misconception is inferred based on common student errors when evaluating limits from graphs with jump discontinuities; it is not explicitly stated in the video.
Students might incorrectly assume that the limit as x approaches 3 is 1, because the solid dot at (3, 1) indicates that f(3) = 1.
The limit describes the value the function approaches as x gets infinitely close to 3, not the value of the function exactly at x=3. For a left-sided limit, we only look at the behavior of the curve for x < 3, which approaches the open circle at y=4.
The graph contains both an open circle at (3,4) and a solid circle at (3,1), so the function value at x=3 is not the same kind of object as the one-sided limiting values.
The spoken explanation in this interval discusses Do not confuse f(3) with the two-sided limit.
This caution is an analyst-added clarification; the video does not explicitly name this misconception.
A learner may think the value of the function at x=3 automatically determines lim_{x -> 3} f(x).
The two-sided limit depends on what values f(x) approaches from both sides, not just the single plotted value at x=3. Here the graph shows different approached heights from left and right, so the two-sided limit does not exist regardless of the filled point at (3,1).
The board distinguishes \lim_{x -> 3^-} f(x)=4 and \lim_{x -> 3^+} f(x)=1 from any direct statement about f(3).
The open circle at (3,4) and solid circle at (3,1) show endpoint inclusion/exclusion visually.
This is an analyst-added reminder, not a separately spoken warning in the clip.
A learner may treat the filled or open dot itself as the limit without considering the direction of approach.
Each one-sided limit is determined by the branch being followed as x approaches 3. The open circle at (3,4) supports the left-hand limit 4, while the solid circle at (3,1) supports the right-hand limit 1.
The spoken explanation in this interval discusses the displayed limit step.
The red dot tracing the curve directly links the geometric path to the algebraic concept of approaching a value.
The method of graphical evaluation is applied to determine the specific value of the left-sided limit defined by the notation.
The spoken explanation in this interval discusses the displayed limit step.
The bottom equation is completed before the top equation is annotated 'does not exist'.
The right-hand limit value is needed before the existence criterion for the two-sided limit can be applied.
The visual break at x=3 with different approached heights is the central picture used throughout the clip.
The spoken explanation in this interval discusses the displayed limit step.
The graph provides the concrete case to which the general two-sided-limit existence rule is applied.
The board places lim_{x -> 3} f(x) above the two one-sided statements and then marks the top one nonexistent.
The spoken explanation in this interval discusses the displayed limit step.
The two-sided limit is distinguished from each one-sided limit by requiring agreement of both directional approaches.
The spoken explanation in this interval discusses the displayed limit step.
The notation is written and solved.
The graph clearly displays both an open circle at (3, 4) and a closed circle at (3, 1), and the calculated limit is 4.
The spoken explanation in this interval discusses the displayed limit step.
The result is written as \lim_{x \to 3^+} f(x)=1.
The spoken explanation in this interval discusses the displayed limit step.
The displayed one-sided limits are 4 and 1, and the top limit is marked 'does not exist'.
The graph shows an open circle at (3,4) and a solid circle at (3,1).
These correspond respectively to the left-hand and right-hand limiting values written on the board.
The graph includes a filled point at (3,1) while the left branch approaches (3,4) with an open endpoint.
The spoken explanation in this interval discusses the displayed limit step.
This search target reflects a common learner question supported by the visuals, though the video does not phrase it as a separate FAQ.
Covered · Introduction to the problem and setting up the notation for the left-sided limit.
Covered · Explanation of the left-sided limit notation and the strategy for evaluating it graphically.
Covered · Step-by-step visual and verbal evaluation of the left-sided limit using the graph, concluding with the final answer.
Covered · The clip introduces the graph, samples points on the right branch, and formally writes the right-hand limit as 1.
Covered · The speaker compares the one-sided limits, states the existence criterion, and annotates the two-sided limit as nonexistent.
Candidate from reviewed en material v1: For an ordinary finite two-sided limit on a real interval around the target, both finite one-sided limits must exist and equal the same number. Agreement gives the two-sided limit; disagreement rules it out. This exercise has unequal finite values.
Candidate from reviewed zh material v1: 对于目标附近实数区间上的通常有限双侧极限，两侧的有限极限必须存在且等于同一个数。相等时得到双侧极限；不相等时双侧极限不存在。本题两侧有限值不同。
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