limit of the multivariable function (KristaKingMath)

1. Formula
   Observation

   The board shows $(x,y)\to(0,0)$ in the limit expression.

2. Audio
   Observation

   The speaker says the problem asks for the limit as $x,y$ approaches zero zero.

(x,y)

An ordered pair of real variables representing a point in the plane from which the function is evaluated.

Points in $\mathbb{R}^2$, especially near $(0,0)$.

1. Formula
   Observation

   The limit notation contains $(0,0)$.

2. Diagram
   Observation

   A green dot marks the origin on the coordinate sketch.

(0,0)

The target point in the plane toward which $(x,y)$ is taken to approach.

A specific point in $\mathbb{R}^2$.

1. Formula
   Observation

   The numerator written on the board is $x^4-4y^2$.

2. Audio
   Observation

   The speaker reads it as x to the fourth minus four y squared.

$x^4-4y^2$

The numerator of the rational two-variable function whose limit is being studied.

Defined for all real $x$ and $y$.

1. Formula
   Observation

   The denominator written on the board is $x^2+2y^2$.

2. Audio
   Observation

   The speaker reads it as quantity x squared plus two y squared.

$x^2+2y^2$

The denominator of the rational two-variable function whose limit is being studied.

Defined for all real $x$ and $y$; it equals $0$ only at $(0,0)$.

1. Formula
   Observation

   The displayed expression is $\lim_{(x,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$.

1. The video does not explicitly introduce a separate name such as $f(x,y)$; this label is an analyst convenience.

$f(x,y)=\frac{x^4-4y^2}{x^2+2y^2}$

The multivariable rational function under consideration in the limit problem.

All $(x,y)\in\mathbb{R}^2$ with $(x,y)\neq(0,0)$.

1. Audio
   Observation

   The speaker mentions approaching along the y-axis.

2. Animation
   Observation

   A yellow arrow is drawn down the vertical axis toward the origin.

$x=0$

The vertical-axis path used as one possible way to approach $(0,0)$.

Points of the form $(0,y)$ with $y\to 0$.

1. Audio
   Observation

   The speaker mentions approaching along the x-axis.

2. Animation
   Observation

   A yellow arrow is drawn along the horizontal axis toward the origin.

$y=0$

The horizontal-axis path used as one possible way to approach $(0,0)$.

Points of the form $(x,0)$ with $x\to 0$.

1. Audio
   Observation

   The speaker says we could look at the value as we approach along the line y equals x.

2. Animation
   Observation

   A diagonal yellow arrow is drawn toward the origin.

$y=x$

A diagonal straight-line path through the origin used to test directional behavior.

Points of the form $(t,t)$ with $t\to 0$.

1. Audio
   Observation

   The speaker says we can try some different lines too, or just along some line y equals mx like this one that we drew here.

2. Animation
   Observation

   Additional yellow arrows are drawn from other directions toward the origin.

1. The symbol $m$ is spoken but not written on the board.

y=mx

A family of straight-line paths through the origin with slope $m$, used to search for different limiting values.

Points of the form $(t,mt)$ with $t\to 0$ and fixed real slope $m$.

1. Audio
   Observation

   The speaker says we really have to show that as we come in here on the line y equals x squared.

2. Animation
   Observation

   A curved yellow path is drawn into the origin, visually distinct from the straight arrows.

1. The equation $y=x^2$ is spoken rather than written on the board.

$y=x^2$

A curved path through the origin mentioned as another possible approach direction.

Points of the form $(t,t^2)$ with $t\to 0$.

1. Formula
   Observation

   The variable x appears in the displayed function and in the path notation (x,0).

x

First coordinate variable of the two-variable function.

Real values; in the worked paths it is either fixed at 0 or allowed to approach 0.

1. Formula
   Observation

   The variable y appears in the displayed function and in the path notation (0,y).

y

Second coordinate variable of the two-variable function.

Real values; in the worked paths it is either fixed at 0 or allowed to approach 0.

1. Formula
   Observation

   The board displays: Find the limit of the multivariable function, or state that it DNE. $\lim_{(x,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$.

2. Audio
   Observation

   The speaker introduces the task as finding the limit of a multivariable function or stating that it does not exist.

The clip sets up a concrete limit question for a rational function of two variables at the origin. The mathematical object under study is the behavior of $\frac{x^4-4y^2}{x^2+2y^2}$ as $(x,y)$ approaches $(0,0)$ from the plane.

1. The limit point is $(0,0)$.

2. The function is a quotient of polynomial expressions in $x$ and $y$.

3. The denominator vanishes at $(0,0)$, so direct substitution is not immediately informative.

1. Audio
   Observation

   The speaker says that because we have a multivariable function, we have to look at the limit as we approach the point zero zero from different directions.

2. Animation
   Observation

   Multiple yellow arrows are drawn from different directions into the origin on a coordinate sketch.

The video explains a standard strategy for two-variable limits: compare the values obtained when approaching the target point along different paths. If two paths give different limiting values, then the overall limit cannot exist. This mirrors the one-variable idea of checking left-hand and right-hand limits, but in two dimensions there are infinitely many possible directions and curves of approach.

1. Applies to limits of functions of two or more variables.

2. Finding equal values along several paths does not by itself prove existence of the full limit.

3. Finding unequal values along two paths is enough to conclude the full limit does not exist.

1. Problem statement for a two-variable limit

1. Audio
   Observation

   The speaker says we can have limits in specific directions, but the general limit doesn't exist.

2. Audio
   Observation

   The speaker adds that the general limit can only exist if the function approaches the same value no matter which direction we approach from.

The clip distinguishes between limits taken along particular paths and the full two-variable limit. A function may have well-defined behavior along chosen lines or curves while still failing to have a single overall limit at the point, because the general limit requires agreement across all approaches.

1. The general limit requires the same limiting value for every approach to the point.

2. Specific directional or pathwise limits may exist even when the general limit does not.

1. Path-based method for testing multivariable limits

1. Audio
   Observation

   The speaker says the only way we can really do that is by using the precise definition of the limit to show it.

2. Audio
   Observation

   The speaker also says that if many paths keep giving the same answer, then we can go to the precise definition of the limit, which is more complicated, to really prove that it approaches the same value no matter where we approach from.

The video states that path testing is useful for disproving existence, but proving existence of a multivariable limit requires the precise definition of the limit. In other words, checking finitely many paths is not enough to establish the full limit; one must control the function's behavior for all sufficiently close points.

1. Used when path tests do not produce a contradiction.

2. Required to prove the general limit exists rather than merely to disprove it.

1. Path-based method for testing multivariable limits
2. Directional limits versus the general limit

1. Formula
   Observation

   Top text reads: Find the limit of the multivariable function, or state that it DNE.

2. Formula
   Observation

   Displayed limit: $\lim_{(x,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$.

The clip presents a standard multivariable calculus problem: determine whether the limit of $f(x,y)=\frac{x^4-4y^2}{x^2+2y^2}$ exists as (x,y) approaches (0,0), or conclude that it does not exist.

1. The target point is the origin (0,0).

2. The function is a quotient of polynomial expressions in x and y.

1. Audio
   Observation

   The speaker says that approaching along the y-axis means $x=0$ everywhere on that line.

2. Formula
   Observation

   Board writes $\lim_{(0,y)\to(0,0)} \frac{-4y^2}{2y^2}=-2$.

To test the two-variable limit, the instructor first restricts the approach to the y-axis by setting $x=0$. This converts the original function into a one-variable expression in y, which can then be simplified and evaluated as y approaches 0.

1. Along the chosen path, x is fixed at 0.

2. The simplification uses cancellation of $y^2$ for $y\neq 0$ before taking the limit.

1. Problem statement: limit of a two-variable rational function at the origin

1. Audio
   Observation

   The speaker asks what happens if we approach along the x-axis and states that $y=0$ everywhere on that axis.

2. Formula
   Observation

   Board writes $\lim_{(x,0)\to(0,0)} \frac{x^4}{x^2}=x^2=0^2=0$.

The second test path is the x-axis, obtained by setting $y=0$. The function reduces to a one-variable expression in x, which simplifies to $x^2$ and then evaluates to 0 as x approaches 0.

1. Along the chosen path, y is fixed at 0.

2. After simplification, the remaining expression still depends on x, so the endpoint value $x=0$ is substituted.

1. Problem statement: limit of a two-variable rational function at the origin

1. Audio
   Observation

   The speaker summarizes that the y-axis gives -2 and the x-axis gives 0, so because these are different values, the limit at (0,0) does not exist.

2. Formula
   Observation

   Board writes -2 $\neq 0$ and boxes DNE.

The clip uses the two-path test: if a multivariable function approaches different values along two different paths to the same point, then the overall limit at that point cannot exist. Here the two computed path limits are -2 and 0, so the final answer is DNE.

1. Both paths must approach the same point (0,0).

2. The two path limits must be unequal.

1. Path restriction to the y-axis
2. Path restriction to the x-axis

1. Audio
   Observation

   The speaker explains approaching the origin along different lines to see if the value is different.

2. Formula
   Observation

   Two different limits are calculated: one equals -2 and the other equals 0.

To show that a multivariable limit does not exist, one can approach the target point along two different paths. If the function yields different limiting values along these paths, the overall limit does not exist.

1. The paths must both approach the same target point.

2. The limits along the chosen paths must exist.

1. Audio
   Observation

   The speaker states that if all paths yielded the same value, one would need to use the precise definition of the limit to prove it exists because there are an infinite number of paths.

The formal epsilon-delta definition required to rigorously prove that a limit exists, as testing a finite number of paths is insufficient for a general proof.

1. Used when attempting to prove a limit exists, not to disprove it.

1. Audio
   Observation

   The speaker says that if the value along the y-axis is different from the value along the line y equals x, then we know that we can approach this point zero zero from two directions and get two different values, which tells us that the general limit does not exist.

If a two-variable function approaches two different limiting values along two different paths ending at the same point, then the general limit at that point does not exist.

1. There are two distinct paths approaching the same point.

2. The function has a limiting value along each path.

3. Those two limiting values are different.

For a function of two variables and a fixed target point, existence of two paths with unequal path-limits implies nonexistence of the full limit.

1. Audio
   Observation

   The speaker says the general limit can only exist if the function approaches the same value no matter which direction we approach from.

A general multivariable limit at a point can exist only if the function approaches the same value along every possible approach to that point.

1. The function is being considered near a point in the plane.

2. One is asking about the full two-variable limit rather than a single path limit.

For all approaches to the point, the limiting value must be the same.

1. Audio
   Observation

   The speaker says that if we keep getting the same answer every time, then we can go to the precise definition of the limit ... to really prove that it approaches the same value no matter where we approach from.

Obtaining the same limiting value along several tested paths does not by itself prove that the full multivariable limit exists; the precise definition is needed for a rigorous existence proof.

1. Several paths have been tested and produced the same limiting value.

2. The goal is to prove existence of the general limit, not merely to search for a counterexample.

For any finite collection of tested paths, matching path-limits do not imply the unrestricted limit exists.

1. Audio
   Observation

   The speaker states that because the function approaches two different values at (0,0), the limit does not exist.

2. Formula
   Observation

   The board records -2 $\neq 0$ and then DNE as the final conclusion.

If a function of two variables approaches different limiting values along two different paths to the same point, then the limit at that point does not exist.

1. The function is evaluated near a common point, here (0,0).

2. At least two distinct approach paths are considered.

3. The pathwise limits exist and are unequal.

For a given point and two paths approaching that point.

1. Audio
   Observation

   The speaker concludes 'we know that this particular limit does not exist.'

2. Formula
   Observation

   The board shows '-2 $\neq 0$' and a boxed 'DNE'.

The limit $\lim_{(x,y)\to(0,0)} \frac{x^4 - 4y^2}{x^2 + 2y^2}$ does not exist.

1. The function approaches -2 along the y-axis.

2. The function approaches 0 along the x-axis.

3. -2 is not equal to 0.

$N/A$

1. Audio
   Observation

   The speaker says finding the limit here is going to be similar to the process we used to find the limit when we were dealing with a two-dimensional function, and recalls checking the left-hand limit and the right-hand limit.

2. Audio
   Observation

   The speaker then says that for a multivariable function we have to look at the limit as we approach the point from different directions.

1. The phrase 'two-dimensional function' in the audio appears to refer to the earlier one-variable setting being contrasted with the present multivariable setting; the exact wording is preserved from the source.

1. Expression

   $$$$
   Explanation

   In the simpler single-variable setting, the speaker recalls checking the left-hand limit and the right-hand limit.

   Justification

   Stated directly in the audio as prior knowledge being reviewed.

   Shown in the video
2. Expression

   $$$$
   Explanation

   If those one-sided limits both existed and were equal, then the general limit existed.

   Justification

   This is the familiar criterion for ordinary one-variable limits.

   Shown in the video
3. Expression

   $$$$
   Explanation

   For a function of two variables, the analogous idea is to compare approaches from different directions in the plane.

   Justification

   The speaker explicitly says we follow the same process except that now we approach from different directions.

   Shown in the video

The clip motivates the path-testing method by analogy with one-sided limits in single-variable calculus.

1. Audio
   Observation

   The speaker says typically we start by approaching along the y-axis and along the x-axis.

2. Audio
   Observation

   The speaker continues that if we still get the same answers there, then we'll try approaching along the line y equals x or just along some line y equals mx.

3. Audio
   Observation

   The speaker adds that if those also give the same answers, we can try some different lines too, and if we keep getting the same answer every time, then we can go to the precise definition of the limit.

1. Expression

   $$$$
   Explanation

   First test the coordinate-axis paths, namely the $x$-axis and $y$-axis.

   Justification

   These are the simplest paths through the origin and are explicitly named first by the speaker.

   Shown in the video
2. Expression

   $$$$
   Explanation

   If those agree, test a diagonal line such as $y=x$ or a more general line $y=mx$.

   Justification

   The speaker says this is the next step when the axes give the same answer.

   Shown in the video
3. Expression

   $$$$
   Explanation

   If those also agree, continue trying other lines.

   Justification

   The speaker says we can try some different lines too.

   Shown in the video
4. Expression

   $$$$
   Explanation

   Only if repeated path tests keep producing the same value should one switch to the precise definition to attempt a proof of existence.

   Justification

   This is stated directly in the audio.

   Shown in the video

The practical workflow in the clip is to search for a counterexample path first, escalating from axes to slanted lines and then to other curves before considering a rigorous existence proof.

1. Audio
   Observation

   The speaker explains that along the y-axis $x=0$, then substitutes 0 for x in the function.

2. Formula
   Observation

   The board shows the substitution and simplification to -2.

1. Expression

   $$\lim_{(0,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$$
   Explanation

   Start from the original limit and restrict the approach to points of the form (0,y).

   Justification

   This is the chosen test path: the y-axis.

   Shown in the video
2. Expression

   $$\lim_{(0,y)\to(0,0)} \frac{0^4-4y^2}{0^2+2y^2}$$
   Explanation

   Substitute $x=0$ everywhere in the numerator and denominator.

   Justification

   Every point on the y-axis has x-coordinate 0.

   Shown in the video
3. Expression

   $$\lim_{(0,y)\to(0,0)} \frac{-4y^2}{2y^2}$$
   Explanation

   Simplify $0^4$ and $0^2$ to 0, leaving only the $y^2$ terms.

   Justification

   Arithmetic simplification of constants.

   Shown in the video
4. Expression

   $$-2$$
   Explanation

   Cancel the common factor $y^2$ and divide -4 by 2.

   Justification

   For $y\neq 0$ the quotient equals -2, so the path limit is -2.

   Shown in the video

Along the y-axis, the function approaches -2.

1. Audio
   Observation

   The speaker says that along the x-axis $y=0$, then substitutes 0 for y and simplifies.

2. Formula
   Observation

   The board shows $\frac{x^4}{x^2}=x^2=0^2=0$.

1. Expression

   $$\lim_{(x,0)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$$
   Explanation

   Start from the original limit and restrict the approach to points of the form (x,0).

   Justification

   This is the chosen test path: the x-axis.

   Shown in the video
2. Expression

   $$\lim_{(x,0)\to(0,0)} \frac{x^4-4(0)^2}{x^2+2(0)^2}$$
   Explanation

   Substitute $y=0$ everywhere in the numerator and denominator.

   Justification

   Every point on the x-axis has y-coordinate 0.

   Shown in the video
3. Expression

   $$\lim_{(x,0)\to(0,0)} \frac{x^4}{x^2}$$
   Explanation

   The terms containing y vanish, leaving $x^4$ over $x^2$.

   Justification

   Arithmetic simplification after substitution.

   Shown in the video
4. Expression

   $$\lim_{(x,0)\to(0,0)} x^2$$
   Explanation

   Reduce the quotient by canceling $x^2$.

   Justification

   For $x\neq 0$, $\frac{x^4}{x^2}=x^2$.

   Shown in the video
5. Expression

   $$0^2=0$$
   Explanation

   Now let x approach 0 and evaluate the remaining expression.

   Justification

   Direct substitution into the simplified one-variable expression.

   Shown in the video

Along the x-axis, the function approaches 0.

1. Audio
   Observation

   The speaker explicitly compares the two results and concludes the limit does not exist.

2. Formula
   Observation

   The board writes -2 $\neq 0$ and boxes DNE.

1. Expression

   $$-2$$
   Explanation

   Recall the limit value found along the y-axis.

   Justification

   Previously derived path result.

   Shown in the video
2. Expression

   $$0$$
   Explanation

   Recall the limit value found along the x-axis.

   Justification

   Previously derived path result.

   Shown in the video
3. Expression

   $$-2\neq 0$$
   Explanation

   Compare the two path limits.

   Justification

   They are unequal numbers.

   Shown in the video
4. Expression

   $$\text{DNE}$$
   Explanation

   Conclude that the full limit at (0,0) does not exist.

   Justification

   By the two-path criterion for nonexistence of multivariable limits.

   Shown in the video

Because two different approach paths give different limiting values, the overall limit at (0,0) does not exist.

1. Formula
   Observation

   The calculation $\lim_{(0,y)\to(0,0)} \frac{-4y^2}{2y^2} = -2$ is shown on the board.

1. Expression

   $$\lim_{(0,y)\to(0,0)} \frac{x^4 - 4y^2}{x^2 + 2y^2}$$
   Explanation

   Start with the original limit expression.

   Justification

   Problem statement.

   Shown in the video
2. Expression

   $$\lim_{(0,y)\to(0,0)} \frac{0^4 - 4y^2}{0^2 + 2y^2}$$
   Explanation

   Substitute $x = 0$ to approach along the y-axis.

   Justification

   Definition of approaching along the y-axis.

   Derived from the video
3. Expression

   $$\lim_{(0,y)\to(0,0)} \frac{-4y^2}{2y^2}$$
   Explanation

   Simplify the numerator and denominator.

   Justification

   Arithmetic simplification.

   Shown in the video
4. Expression

   $$-2$$
   Explanation

   Cancel the $y^2$ terms (for $y \neq 0$) and evaluate the constant limit.

   Justification

   Algebraic cancellation and limit of a constant.

   Shown in the video

The limit along the y-axis is -2.

1. Formula
   Observation

   The calculation $\lim_{(x,0)\to(0,0)} \frac{x^4}{x^2} = x^2 = 0^2 = 0$ is shown on the board.

1. Expression

   $$\lim_{(x,0)\to(0,0)} \frac{x^4 - 4y^2}{x^2 + 2y^2}$$
   Explanation

   Start with the original limit expression.

   Justification

   Problem statement.

   Shown in the video
2. Expression

   $$\lim_{(x,0)\to(0,0)} \frac{x^4 - 4(0)^2}{x^2 + 2(0)^2}$$
   Explanation

   Substitute $y = 0$ to approach along the x-axis.

   Justification

   Definition of approaching along the x-axis.

   Derived from the video
3. Expression

   $$\lim_{(x,0)\to(0,0)} \frac{x^4}{x^2}$$
   Explanation

   Simplify the expression.

   Justification

   Arithmetic simplification.

   Shown in the video
4. Expression

   $$\lim_{(x,0)\to(0,0)} x^2$$
   Explanation

   Cancel $x^2$ from the numerator and denominator (for $x \neq 0$).

   Justification

   Algebraic cancellation.

   Shown in the video
5. Expression

   $$0^2 = 0$$
   Explanation

   Evaluate the limit as x approaches 0.

   Justification

   Direct substitution.

   Shown in the video

The limit along the x-axis is 0.

1. Formula
   Observation

   The board shows $\lim_{(x,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$ under the instruction to find the limit or state that it DNE.

2. Audio
   Observation

   The speaker describes this as the particular problem and reads the expression aloud.

1. No numerical path evaluations are completed within this 180-second excerpt.

Find $\lim_{(x,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$, or state that the limit does not exist.

1. The function is $\frac{x^4-4y^2}{x^2+2y^2}$.

2. The approach point is $(0,0)$.

3. The task allows either computing the limit or concluding that it does not exist.

Determine whether the two-variable limit exists at the origin, and if not, justify that it does not exist.

1. Expression

   $$$$
   Explanation

   Set up the problem as a multivariable limit at the origin.

   Justification

   Directly shown on the board and read aloud.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Explain that the method will be to compare values obtained by approaching $(0,0)$ along different paths.

   Justification

   Stated in the audio during the conceptual introduction.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Sketch the coordinate plane and mark the origin, then indicate candidate paths such as the axes, $y=x$, $y=mx$, and $y=x^2$.

   Justification

   Shown through the drawing animation and described in the audio.

   Shown in the video
4. Expression

   $$$$
   Explanation

   State the plan to look for two paths that yield different values, which would prove nonexistence.

   Justification

   Explicitly stated by the speaker.

   Shown in the video

Within this excerpt, the final numerical answer is not reached; the clip only establishes the setup and strategy.

The absence of a completed computation is evident from the audio ending mid-explanation and from the board never showing substituted path results.

1. Formula
   Observation

   The entire clip works the displayed example $\lim_{(x,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$.

2. Audio
   Observation

   The speaker narrates each path computation and the final conclusion.

Find $\lim_{(x,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$, or state that it does not exist.

1. The function is $f(x,y)=\frac{x^4-4y^2}{x^2+2y^2}$.

2. The target point is (0,0).

3. Two test paths are used: the y-axis and the x-axis.

Determine whether the two-variable limit exists at (0,0).

1. Expression

   $$\lim_{(0,y)\to(0,0)} \frac{-4y^2}{2y^2}=-2$$
   Explanation

   Approach along the y-axis by setting $x=0$ and simplify.

   Justification

   Path restriction plus algebraic cancellation.

   Shown in the video
2. Expression

   $$\lim_{(x,0)\to(0,0)} \frac{x^4}{x^2}=\lim_{(x,0)\to(0,0)} x^2=0$$
   Explanation

   Approach along the x-axis by setting $y=0$ and simplify.

   Justification

   Path restriction, cancellation, then substitution $x=0$.

   Shown in the video
3. Expression

   $$-2\neq 0$$
   Explanation

   Compare the two path limits.

   Justification

   Different numerical results along two valid paths to the same point.

   Shown in the video
4. Expression

   $$\text{DNE}$$
   Explanation

   State the final conclusion for the original two-variable limit.

   Justification

   Two-path test for nonexistence.

   Shown in the video

The limit does not exist (DNE).

The conclusion is verified by exhibiting two explicit paths to (0,0) that yield different limiting values, -2 and 0.

1. Formula
   Observation

   The entire problem and solution are written on the board.

2. Audio
   Observation

   The speaker narrates the process of finding the limit or stating it DNE.

Find the limit of the multivariable function $\lim_{(x,y)\to(0,0)} \frac{x^4 - 4y^2}{x^2 + 2y^2}$, or state that it DNE.

1. The function $f(x,y) = \frac{x^4 - 4y^2}{x^2 + 2y^2}$

2. The target point $(0,0)$

Determine if the limit exists and find its value, or prove it does not exist.

1. Expression

   $$\lim_{(0,y)\to(0,0)} \frac{-4y^2}{2y^2} = -2$$
   Explanation

   Approach the origin along the y-axis (set $x=0$) and evaluate the limit.

   Justification

   Two-path test method.

   Shown in the video
2. Expression

   $$\lim_{(x,0)\to(0,0)} \frac{x^4}{x^2} = 0$$
   Explanation

   Approach the origin along the x-axis (set $y=0$) and evaluate the limit.

   Justification

   Two-path test method.

   Shown in the video
3. Expression

   $$-2 \neq 0$$
   Explanation

   Compare the results from the two different paths.

   Justification

   Logical comparison.

   Shown in the video
4. Expression

   $$\text{DNE}$$
   Explanation

   Conclude that the overall limit does not exist.

   Justification

   Two-path test theorem.

   Shown in the video

The limit does not exist (DNE).

The two chosen paths yield different limiting values (-2 and 0), which is sufficient to prove non-existence.

1. Diagram
   Observation

   A black chalkboard fills the frame with white handwritten text at the top and the limit expression centered below it.

2. Formula
   Observation

   The visible formula is $\lim_{(x,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$.

1. White handwritten instruction line

2. Limit expression

3. Blackboard background

1. No new writing appears during most of this interval.

2. The cursor or pointer moves around the formula while the speaker reads and explains it.

1. The problem statement remains unchanged on screen.

2. The target point $(0,0)$ and the rational expression stay visible throughout.

This visual phase anchors the mathematical task before any geometric explanation begins.

1. Animation
   Observation

   A vertical double-headed arrow is drawn first, then a horizontal double-headed arrow crossing it, forming coordinate axes.

2. Animation
   Observation

   A green dot is placed at the intersection of the axes.

1. Vertical axis

2. Horizontal axis

3. Green dot at the intersection

1. The blank lower-left portion of the board becomes a simple coordinate sketch.

2. The origin is highlighted with a colored point.

1. The original limit expression remains at the top of the board.

2. The axes represent the standard $x$-$y$ plane around $(0,0)$.

The sketch converts the abstract limit point $(0,0)$ into a geometric location that can be approached from many directions.

1. Animation
   Observation

   A yellow rectangular highlight appears around $(0,0)$ in the limit expression.

1. Yellow box around $(0,0)$

2. Limit expression

1. The target point in the formula is visually singled out.

1. The rest of the formula is unchanged.

2. The coordinate sketch remains visible below.

This links the algebraic notation $(x,y)\to(0,0)$ to the geometric origin just drawn.

1. Animation
   Observation

   Yellow arrows are drawn sequentially toward the origin along the vertical axis, horizontal axis, and a diagonal direction.

2. Animation
   Observation

   Later, additional yellow arrows and a curved yellow path are added, indicating more ways to approach the origin.

3. Audio
   Observation

   The speaker names the y-axis, x-axis, the line $y=x$, lines of the form $y=mx$, and the curve $y=x^2$ while these marks are being made.

1. Some later arrows are generic illustrations of other directions rather than labeled equations written on the board.

1. Yellow arrow down the y-axis

2. Yellow arrow along the x-axis

3. Yellow diagonal arrow for $y=x$

4. Additional yellow arrows for other lines

5. Curved yellow path associated with $y=x^2$

1. The number of indicated approach directions increases over time.

2. The sketch evolves from two axes into a multi-path diagram around the origin.

1. All drawn paths terminate at the same green origin point.

2. The underlying function and limit statement do not change.

The animation visualizes the central idea that a two-variable limit depends on behavior under many possible approaches, not just one.

1. Diagram
   Observation

   A blackboard shows the problem text at top, the limit expression near the upper middle, and a coordinate sketch on the left with arrows pointing toward the origin.

1. Top instruction text

2. Limit expression $\lim_{(x,y)\to(0,0)} \frac{x^4-4y^2}{x^2+2y^2}$

3. Left-side coordinate axes sketch

4. Multiple arrows directed toward the origin

1. New handwritten work is added below the original problem as the solution progresses.

2. The final boxed DNE is written near the end.

1. The original problem statement remains visible throughout.

2. The coordinate sketch continues to represent approach toward the origin.

The visual organization separates the problem statement from the step-by-step path calculations, while the arrow diagram reinforces that the issue is the behavior of the function as (x,y) approaches (0,0) from different directions.

1. Diagram
   Observation

   The board adds $\lim_{(0,y)\to(0,0)}$ and then the simplified fraction leading to -2.

1. Expression $\lim_{(0,y)\to(0,0)}$

2. Fraction $\frac{-4y^2}{2y^2}$

3. Result -2

1. The path notation (0,y) is introduced.

2. The substituted fraction is written and simplified to -2.

1. The original function at the top remains unchanged.

This visual step encodes the restriction $x=0$ and shows the resulting one-variable limit along the y-axis.

1. Diagram
   Observation

   The board adds $\lim_{(x,0)\to(0,0)}$ and the chain $\frac{x^4}{x^2}=x^2=0^2=0$.

1. Expression $\lim_{(x,0)\to(0,0)}$

2. Fraction $\frac{x^4}{x^2}$

3. Simplified expression $x^2$

4. Result 0

1. The path notation (x,0) is introduced.

2. The substituted fraction is simplified to $x^2$ and then evaluated at 0.

1. The earlier y-axis result -2 remains visible above.

This visual step encodes the restriction $y=0$ and shows the resulting one-variable limit along the x-axis.

1. Diagram
   Observation

   The board writes -2 $\neq 0$ and then boxes DNE.

1. Inequality -2 $\neq 0$

2. Boxed label DNE

1. The inequality comparing the two path limits is written.

2. The final answer DNE is boxed.

1. Both prior path results remain visible.

The visual emphasis on the boxed DNE marks the logical endpoint of the two-path argument.

1. Diagram
   Observation

   The blackboard displays the problem statement at the top, a coordinate system with colored arrows indicating paths on the left, and the step-by-step calculations below.

1. Coordinate axes

2. Colored arrows representing paths

3. Mathematical expressions

4. Boxed 'DNE' conclusion

1. The speaker points to different parts of the board while explaining the concepts.

1. The written mathematical content remains static throughout the clip.

The visual layout organizes the problem, the geometric intuition (paths), and the algebraic execution of the two-path test.

1. Audio
   Observation

   The speaker says that if we keep getting the same answer every time, then we can go to the precise definition of the limit ... to really prove that it approaches the same value no matter where we approach from.

One might think that if several tested paths all give the same limiting value, then the full multivariable limit must exist.

The video explicitly warns against this: path testing can help find nonexistence, but proving existence requires the precise definition of the limit because all approaches must be controlled.

1. Audio
   Observation

   The speaker says we can have limits in specific directions, but the general limit doesn't exist.

One might assume that a limit along a particular direction is enough to define the overall limit at the point.

The clip distinguishes pathwise or directional behavior from the general two-variable limit, which requires the same value for every approach to the point.

1. Audio
   Observation

   The speaker says that if both paths had given -2, then one might have wanted to try a couple other values as well.

Getting the same limit along two chosen paths is enough to conclude that the full multivariable limit exists.

The video indicates that equal results on two paths would still leave room for testing additional paths; agreement on finitely many paths does not establish existence.

1. Audio
   Observation

   The speaker explicitly warns that even if many paths gave the same value, it wouldn't prove the limit exists without the precise definition, because there are an infinite number of paths.

Believing that if a function approaches the same value along several tested paths, the multivariable limit must exist.

Testing a finite number of paths can only disprove the existence of a limit. To prove it exists, one must use the precise epsilon-delta definition to account for all infinite possible paths.

1. Audio
   Observation

   The speaker compares the present multivariable problem to the earlier process of checking left-hand and right-hand limits.

The method of comparing approaches is presented as an extension of the familiar one-variable idea of matching one-sided limits to the two-variable setting.

1. Audio
   Observation

   The speaker explains that two different path values imply the general limit does not exist.

The path-testing method relies on the proposition that disagreement between two path-limits is sufficient to conclude nonexistence of the full limit.

1. Audio
   Observation

   The speaker contrasts limits in specific directions with the general limit.

The clip uses the distinction between directional behavior and the unrestricted limit to explain why the all-directions condition matters.

1. Audio
   Observation

   The speaker says that after repeated path tests give the same answer, one must use the precise definition of the limit to prove existence.

The precise-definition approach is presented as the stronger, fully rigorous method that supersedes finite path checking when one wants to prove the limit exists.

1. Audio
   Observation

   The speaker moves from the general limit problem to the specific y-axis path.

2. Formula
   Observation

   The board rewrites the original limit using (0,y) -> (0,0).

The general two-variable limit problem is investigated by applying a path restriction to the y-axis.

1. Audio
   Observation

   The speaker then asks what happens along the x-axis.

2. Formula
   Observation

   The board rewrites the original limit using (x,0) -> (0,0).

The same original limit is tested again using a second path restriction, now to the x-axis.

1. Audio
   Observation

   The speaker compares the two results and concludes DNE.

2. Formula
   Observation

   The board writes -2 $\neq 0$ and boxes DNE.

The final nonexistence conclusion depends on the previously computed y-axis path limit.

1. Audio
   Observation

   The speaker compares the two results and concludes DNE.

2. Formula
   Observation

   The board writes -2 $\neq 0$ and boxes DNE.

The final nonexistence conclusion also depends on the previously computed x-axis path limit.

1. Audio
   Observation

   The speaker explicitly contrasts the y-axis result with the x-axis result.

2. Formula
   Observation

   The board displays -2 from one path and 0 from the other, then -2 $\neq 0$.

The two methods are contrasted because they use different coordinate restrictions and produce different limiting values.

1. Audio
   Observation

   The speaker transitions from discussing the results of the path tests to mentioning the need for the precise definition if the values had matched.

The two-path test is used to disprove the existence of a limit by finding contradictory evidence, whereas the precise definition is required to prove the existence of a limit across all possible paths.

1. Audio
   Observation

   The speaker explains that two different path values imply the general limit does not exist.

1. Path-based method for testing multivariable limits
2. Two-path disagreement criterion for nonexistence

1. Audio
   Observation

   The speaker says that if many paths give the same answer, one still needs the precise definition of the limit to prove existence.

1. Role of the precise epsilon-delta style definition
2. Path agreement does not replace the precise proof of existence
3. Mistaking repeated path agreement for a proof of existence

1. Audio
   Observation

   The speaker distinguishes limits in specific directions from the general limit.

1. Directional limits versus the general limit
2. All-directions requirement for a general limit
3. Confusing a directional limit with the general limit

1. Audio
   Observation

   The speaker lists a typical order: y-axis, x-axis, then $y=x$ or y=mx, then other lines.

1. Path-based method for testing multivariable limits
2. Ordered strategy for testing paths

1. Formula
   Observation

   The opening board text and displayed limit define the task.

1. Problem statement: limit of a two-variable rational function at the origin