Limits of Vector Functions

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Definition: If is a vector-valued function, then provided that the limits of the components exist.

//Limits of vector-valued functions in are defined similarly as the limit of each component. //

Let's look at some examples of evaluating limits of vector-valued functions. Consider the vector-valued function and suppose that we wanted to compute . To compute this limit, all we need to do is compute the limits of the components.

For another example, consider the vector-valued function and suppose that we wanted to compute . To compute this limit, we will compute all of the limits of the components again, however, this time the limits are a little trickier to compute. Fortunately, we have already learned about various rules to evaluate limits.

For we will use L'Hospital's Rule, and so .

For , we can use direct substitution and so .

Now is also easy to compute by direct substitution and so .

Thus we have that .

The following theorem gives us a formal definition to say a vector-valued function has limit at , which is analogous to that of limits of real-valued functions.

Theorem 1: Let be a vector-valued function and let . Then if and only if such that if then .
  • Proof: Suppose that . Then we have that:
  • Now recall that two vectors are equal if and only if their components are equal, and so the equation above implies that , , and . Now notice that these three limits are limits of real-valued functions.
  • Since then such that if then .
  • Since then such that if then .
  • Since then such that if then .
  • Let . Then if we have that:
  • Suppose that such that if then . Therefore we have that
, which implies that:


  • Now since all terms of the lefthand side of this equation are positive, we must have that for then , , and , and so , and . Therefore by the definition of real-valued function limits we have that , , and .
  • Thus .

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