Limit of a vector-valued function

 Goal: Determine the limit of a vector-valued function

Definition

This is a rigorous definition. In practice we apply the following theorem:

Theorem

Limit of a vector-valued function

Let f, g, h be functions of t and r(t) = f(t)i + g(t)j, the limit of r as t approaches a is defined by:

Similarly, the limit of the vector-valued function r(t) = f(t)i + g(t)j + h(t)k as t approaches a is defined by:

Examples

For each of the vector-valued functions calculate lim r(t) when t approaches 3:

a. r(t) = (t²- 3t + 4)i + (4t + 3)j

b. r(t) = (2t-4/t+1)i + (t/t²+1)j + (4t - 3)k

Solution

a. Using the definition of limit above and substituting t, we have:

 

b. Doing the same:

Practice

Calculate limr(t) when t approaches 2 for: