SUNY Geneseo Department of Mathematics

Limits of Multivariable Functions

Tuesday, March 7

Math 223
Spring 2023
Prof. Doug Baldwin

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The Math Learning Center closes for break this Thursday (March 9) at 6:00 PM.

It reopens after break on Monday, March 20 at 9:30 AM.

Limits of Multivariable Functions

Based on section 3.2 in the textbook.

Key Ideas

The definition of a multivariable limit. The definition captures the idea that in order for the limit of a multivariable function f(x,y,z,...) to be L as the inputs approach some point P, it must be the case that you can get f to come arbitrarily close (i.e., within tolerance ε) to L by bringing the inputs close enough (within distance δ) of P.

Surface of function above point in X Y plane; within distance delta of point function value is within epsilon of L

Multivariable limit laws are analogous to the single-variable ones.

Recognizing when a limit doesn’t exist is more complicated than in the single-variable case (we’ll talk more about this tomorrow).

The definition of continuity is the same as for single-variable functions, namely f is continuous at point P if f(P) is defined, the limit as the inputs approach P is defined, and the function’s value and the limit are equal.

New vocabulary of interior points, boundary points, open vs closed regions, connected regions, etc. will reappear throughout our later study of multivariable functions.

Limit Examples

Find lim_{(x,y)→(1,2)} (x² + y²) / (x - 2y)

You can just plug in 1 for x and 2 for y and evaluate to find this limit. But remember that “just plug in” is really shorthand for using limit laws over and over (in this case, the quotient law, then addition, power law, etc.)

Finding limit of X squared plus Y squared all over X minus 2 Y by plugging in X and Y values

How about lim_{(x,y,z)→(-2,2,1)} (x² - y²) / z(x + y)

If you plug in x = -2, y = 2, and z = 1 you get an indeterminate form 0/0. Unfortunately, L’Hôpital’s Rule doesn’t apply to multivariable functions, so you have to deal with indeterminate forms by trying to algebraically simplify the expression whose limit you’re taking. In this case, we did that by factoring the numerator and then canceling:

Finding limit of X squared minus Y squared all over Z times X plus Y by factoring and canceling

Continuity Examples

Is f(x,y,z) = x(y - z) / (xy + xz) continuous at (0, 1, 2)?

Checking the requirements for continuity, we find that the limit exists, but the function is undefined at (0, 1, 2), so the function is not continuous there:

A function, its value at a point, and its limit; function is undefined and so isn't continuous at the point

How about at (2, 1, 0)?

Here the function and limit both exist and are equal, so the function is continuous:

A function, its value at a point, and its limit; function value and limit are equal, so function is continuous

Limits that don’t exist, and vocabulary related to regions, their interiors and boundaries, etc.

No new reading.

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