In this section we need to talk briefly about limits, derivatives and integrals of vector functions. As you will see, these behave in a fairly predictable manner. We will be doing all of the work in 
Let’s start with limits. Here is the limit of a vector function.
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So, all that we do is take the limit of each of the component's functions and leave it as a vector.
Example 1 Compute
   
Solution
There really isn’t all that much to do here.
 
Notice that we had to use L’Hospital’s Rule on the y component.
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Now let’s take care of derivatives and after seeing how limits work it shouldn’t be too surprising that we have the following for derivatives.
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Example 2 Compute
   
Solution
There really isn’t too much to this problem other than taking the derivatives.
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Most of the basic facts that we know about derivatives still hold however, just to make it clear here are some facts about derivatives of vector functions.
Facts
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There is also one quick definition that we should get out of the way so that we can use it when we need to.
A smooth curve is any curve for which 
Finally, we need to discuss integrals of vector functions. Using both limits and derivatives as a guide it shouldn’t be too surprising that we also have the following for integration for indefinite integrals
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and the following for definite integrals.
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With the indefinite integrals we put in a constant of integration to make sure that it was clear that the constant in this case needs to be a vector instead of a regular constant.
Also, for the definite integrals we will sometimes write it as follows,
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In other words, we will do the indefinite integral and then do the evaluation of the vector as a whole instead of on a component by component basis.
Example 3 Compute
   
Solution
All we need to do is integrate each of the components and be done with it.
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Example 4 Compute
   
Solution
In this case all that we need to do is reuse the result from the previous example and then do the evaluation.
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