SUNY Geneseo Department of Mathematics

Limits of Multivarible Functions

Thursday, October 11

Math 223 01
Fall 2018
Prof. Doug Baldwin

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Questions?

Problem set question 3?

The question asks where an ant is after it walks 1 unit (let’s suppose the distance units are inches) along a curve r(t), assuming it starts at r(1).

Thinking of this in terms of arc length, the question gives you an arc length and a lower bound for integration, but the upper bound isn’t known. But you can use the arc length integral to find an equation you can solve for the upper bound:

Ant walking distance 1 from r(1) along r(t) means integral from 1 to t of r must be 1

Once you have the upper bound, plug it in as the t value in r(t).

Plotting Multivariable Functions with Mathematica

The ContourPlot function plots level curves. Use ContourPlot3D for level surfaces.

Here are some examples.

Key Points

Mathematica can plot level curves or surfaces

The functions for doing so.

Limits

Section 4.2.

Definitions

δ-Disk

Draw a 2-disk around (1,0). Since a δ-disk is just a circle of radius δ, just draw such a circle centered at (1,0):

Circle with radius 2 around point (1,0)

Other questions? What are δ-disks good for?

They are the 2-variable (and their more general form, δ-balls, are the more-than-2-variable) analogs of the single-variable idea of defining a limit partly in terms of an interval around a certain value of x. The single-variable limit definition requires that you can always keep f(x) as close as you want to a limit value L by keeping x within a certain distance of value a. With 2 variables, that “certain distance” means staying within a small circle around a point (a,b):

2- and 1-variable functions around limit L and corresponding point x = a or (x,y) = (a,b)

Finding Limits

2 Variables

Find lim(x,y)→(1,2)((x+y)/(3xy))2

You can find such limits much as you would with a single variable, namely by plugging in the actual values of x and y (thanks to the limit laws).

Insert a and b values into f(x,y) to find limit as long as f is defined at (a,b)

Key Points

Definition of multivariable limit.

Finding multivariable limits.

Next

When limits of multivariable functions don’t exist.

Then start talking about partial derivatives.

Read section 4.3.

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