SUNY Geneseo Department of Mathematics

Limits of Vector Valued Functions

Tuesday, February 21

Math 223
Spring 2023
Prof. Doug Baldwin

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Limits of Vector-Valued Functions

Based on “Limits and Continuity of a Vector-Valued Function” in section 2.1.

Key Ideas

The rigorous definition of the limit of a vector-valued function: the limit as t approaches a of r(t) is L exactly when the limit of the distance between r(t) and L approaches 0 as t approaches a:

Curve R of T getting closer to point L as T gets closer to A means limit as T goes to A of R of T is L

The theorem for evaluating limits of vector-valued functions: the limit of a vector-valued function is a vector consisting of the limits of the component functions.

Continuity at a point is defined by 3 conditions, which are exactly the same as the conditions for a single-variable function being continuous at a point (the function is defined there, the limit is defined, and the function equals the limit). Thus the same definition of continuity applies in 1, 2, and 3 dimensions (and more).

Questions

Examples of finding limits, especially in cases where the limits don’t exist?

Examples

An example involving limits that don’t exist: consider this limit…

Limit as X goes to 1 of a 3-component vector function

As a first step, take the limits of each component function individually:

Limit of a vector-valued function is a vector of limits of the component functions

The limit in the first component doesn’t exist, partly because it’s infinite, and partly because the limit is a different infinity depending on whether you approach from the left or the right. So this component illustrates infinite limits and the use of one-sided limits to check existence.

2 over 1 minus X has positive and negative infinite limits from left and right respectively

The limit in the second component does exist, as can be seen simply by plugging x = 1 into the second component function.

The limit in the third component also doesn’t exist, because the limits from the left and right are different. But this time both limits are finite.

X minus 1 over X squared minus 1 has different limits from left and right as X goes to 1

Since even one of the component limits doesn’t exist, the limit of the whole vector function doesn’t exist (i.e., we could have stopped after the first component). But continuing the example to the end illustrated use of a lot of techniques from single-variable limits — for instance, factoring to simplify an expression for evaluation, using one-sided limits to deal with absolute values, recognizing infinite limits, etc.

As another example, find limt→1⟨ t - 1, (t2 - 2t + 1) / (t2 - t), sin(πt) / ln t ⟩

This time the component limits all exist, and can be found by plugging t = 1 into the first function, factoring to simplify the second, and using L’Hôpital’s Rule on the third:

Finding a vector-valued limit by finding limits of each component

Next

Derivatives of vector-valued functions.

Please read “Derivatives of Vector-Valued Functions” and “Tangent Vectors and Unit Tangent Vectors” in section 2.2 of the textbook.

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