Source: MATH 163 Abridged Class Notes, Section 14.2
Tags: multivariable limits, path test, limits in 3D, Calculus III, MATH 163, direction-dependent limits, DNE
Difficulty: Intermediate | Prerequisites: Calculus I and II limits (single-variable), familiarity with basic 2D functions
In Calculus I, a limit only needs to agree from the left and the right. In multivariable calculus, you can approach a point from infinitely many directions across the xy-plane, and the limit exists only if every single path gives the same answer. This is the central challenge of Section 14.2, and it changes how you think about limits entirely. If you are comfortable with single-variable limit evaluation, you have the tools you need to start here.
To evaluate limits of multivariable functions, you test what happens along different paths toward the target point. If any two paths give different results, the limit does not exist (DNE). If every path you try gives the same value, the limit might exist, but you need the Squeeze Theorem or polar coordinates to prove it for all paths at once.
Limit of a multivariable function
The value that f(x, y) approaches as the point (x, y) gets arbitrarily close to a target point (a, b), from every possible direction simultaneously.
In simple terms, this means: what number does the output settle on as you zoom in toward (a, b) from all angles at once?
Path (or direction of approach)
A specific curve or line along which you approach the target point, defined by setting a relationship between x and y (e.g. y = 0, x = 0, y = x, y = x^2).
Think of it as: picking one road to walk toward the point, instead of considering every road at once.
DNE (Does Not Exist)
The conclusion reached when two or more paths to the same target point produce different limit values. If the outputs disagree along different directions, no single limit exists.
In simple terms, this means: the function can not make up its mind about what value to approach, because the answer depends on which direction you come from.
Direction-dependent limit
A limit whose value changes based on the path chosen. This is the hallmark of a limit that does not exist in the multivariable setting.
Think of it as: the function gives you a different answer depending on which way you walk toward the point.
In Calculus I, a limit at x = a requires checking only two directions: from the left and from the right. If both agree, the limit exists.
In multivariable calculus, (x, y) can approach (a, b) from infinitely many directions: along the x-axis, the y-axis, the line y = x, the parabola y = x^2, or any other curve through (a, b).
The limit exists only if every possible path of approach gives the same output value.
The idea is to reduce a multivariable limit to a single-variable problem by fixing one variable or setting a relationship between x and y.
Set y = 0 to approach along the x-axis. The function becomes a function of x alone.
Set x = 0 to approach along the y-axis. The function becomes a function of y alone.
Set y = x (or y = mx for a general slope) to approach along a straight line through the origin.
Set y = x^2 or x = y^2 when straight-line paths all agree but you suspect the limit still does not exist.
If any two paths produce different values, the limit is DNE. Full stop. You do not need to check further.
If every path you try produces the same value, that is encouraging but not proof. You would need the Squeeze Theorem or polar coordinates to confirm the limit exists (covered in the companion notes).
Evaluate the limit as (x, y) approaches (0, 0).
Along the x-axis (y = 0): the function becomes x^2 / x^2 = 1, so the limit is 1.
Along the y-axis (x = 0): the function becomes 0 / y^2 = 0, so the limit is 0.
Two paths, two different answers. The limit does not exist.
This is the cleanest possible example of the method: you only needed the two coordinate axes to find disagreement.
Evaluate the limit as (x, y) approaches (0, 0).
Along the x-axis (y = 0): 0 / x^2 = 0. Limit is 0.
Along the y-axis (x = 0): 0 / y^2 = 0. Limit is 0.
Along the line y = x: (y * y) / (y^2 + y^2) = y^2 / (2y^2) = 1/2. Limit is 1/2.
The axes both gave 0, but the diagonal gave 1/2. The limit does not exist.
This example shows why checking only the coordinate axes is not sufficient. Always try at least one diagonal or curved path as well.
This was left as a class exercise. The approach is the same: test along x = 0, y = 0, and y = x (or y = mx). If any two paths disagree, the limit is DNE.
Also left as a class exercise. For this function, all straight-line paths through the origin give 0. The trick is to try the parabola x = y^2, which gives a limit of 1/2. The limit does not exist.
This is a favourite exam problem because it punishes students who only check straight lines.
\lim_{(x,y) \to (0,0)} \frac{x^2}{x^2 + y^2} = \text{DNE}Along y = 0 the value is 1; along x = 0 the value is 0.
\lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^2} = \text{DNE}Along both axes the value is 0; along y = x the value is 1/2.
\lim_{(x,y) \to (0,0)} \frac{xy^2}{x^2 + y^4} = \text{DNE}All straight-line paths give 0; the parabola x = y^2 gives 1/2.
Multivariable limits describe how a system behaves as two (or more) inputs change simultaneously. In engineering, this is how you determine whether a temperature field, pressure distribution, or electromagnetic potential is continuous at a point. If the value depends on the direction of approach, the physical quantity has a discontinuity there, which can signal a phase boundary, a shock wave, or a point where a model breaks down.
Students often think that checking only the x-axis and y-axis is enough. It is not. The function xy / (x^2 + y^2) gives 0 along both axes but 1/2 along y = x.
Students sometimes believe that if every straight line through the origin gives the same limit, the limit must exist. The function xy^2 / (x^2 + y^4) disproves this: every line y = mx gives 0, but the parabola x = y^2 gives 1/2.
A common error is concluding the limit exists after testing a few paths. Path testing can only prove non-existence (by finding disagreement). It cannot prove existence on its own.
Some students confuse "the limit equals 0 along this path" with "the function equals 0 at the point." Limits describe behaviour near the point, not at it.
⚠️ Path testing is tested in nearly every Calc III midterm. Expect at least one "show this limit does not exist" problem.
⚠️ The trickiest exam questions use functions where all straight-line paths agree (e.g. xy^2 / (x^2 + y^4)), forcing you to try a curved path.
⚠️ You will almost certainly be asked to state clearly why the limit does not exist. "Different paths give different values" is the required justification, with the specific values shown.
⚠️ Some exam problems ask you to evaluate a limit (not just show DNE). If path testing gives the same answer on every path you try, switch to the Squeeze Theorem or polar coordinates to confirm.
True or false: If a multivariable limit gives the same value along the x-axis and y-axis, the limit exists. (False. You must check all paths, not just two.)
Fill in the blank: The path-testing method can only prove that a limit ____. (does not exist)
True or false: Setting y = x turns a two-variable limit into a single-variable limit. (True.)
Fill in the blank: For f(x, y) = x^2 / (x^2 + y^2), the limit along y = 0 is ____ and along x = 0 is ____. (1, 0)
True or false: If all straight-line paths give the same limit, no curved path can give a different value. (False. The function xy^2 / (x^2 + y^4) is a counterexample.)
Q: Show that the limit of xy / (x^2 + y^2) as (x, y) approaches (0, 0) does not exist.
A: Along y = 0, the function equals 0/x^2 = 0, so the limit is 0. Along y = x, the function equals x^2 / (2x^2) = 1/2, so the limit is 1/2. Since two paths give different values (0 and 1/2), the limit does not exist.
Q: Evaluate the limit of x^2 / (x^2 + y^2) as (x, y) approaches (0, 0), or show it does not exist.
A: Along y = 0, the function equals 1. Along x = 0, the function equals 0. Two different values, so the limit does not exist.
Q: Why is it not sufficient to check only the coordinate axes when path-testing a multivariable limit?
A: The coordinate axes are only two of infinitely many paths. A function can agree along both axes but disagree along a diagonal or curved path (e.g. xy / (x^2 + y^2) equals 0 on both axes but 1/2 on y = x).
Q: For f(x, y) = xy^2 / (x^2 + y^4), explain why checking y = mx for any constant m is not enough to determine whether the limit exists at the origin.
A: For any line y = mx, substituting gives mx^3 / (x^2 + m^4 x^4) = mx / (1 + m^4 x^2), which approaches 0 as x approaches 0. Every straight line gives 0. But along the parabola x = y^2, the function becomes y^4 / (y^4 + y^4) = 1/2. The limit does not exist, and you can only find this by testing a curved path.
This connects to the Squeeze Theorem and polar coordinates, which are the tools you use when path testing suggests the limit does exist. If every path gives the same value, path testing alone cannot confirm existence, so you switch to those techniques (covered in the companion study notes).
Continuity in multivariable functions builds directly on this material: a function f(x, y) is continuous at (a, b) if the limit equals f(a, b). Understanding when limits fail to exist tells you where continuity breaks down.
Partial derivatives (the next major topic in MATH 163) also rely on limits, specifically limits taken in one direction at a time, which is a restricted version of the full multivariable limit.
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