We can find the graph of  in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

In general:

If ,

then 

Derivative rules that will be needed here:

• Taking a derivative on a term, or using the power rule, can be done by doing the following:
• Special rule when differentiating an exponential function: , where k is a constant.

In this problem, 

Put it all together to get 

Calculate the sum of vectors.

In general,

Solution:

The limit we are given is one sided, meaning we are approaching our x value from one side; in this case, the negative sign exponent indicates that we are approaching 3 from the left side, or using values slightly less than three on approach.

This corresponds to the part of the piecewise function for values less than 3. When we substitute our x value being approached, we get

The limit  indicates that we are trying to find the value of the limit as  approaches to zero from the right side of the graph.  

From right to left approaching , the limit approaches to 1 even though the value at  of the piecewise function does not exist.

The answer is .

Examining the graph of the function above, we need to look at three things:

1) What is the limit of the function as it approaches zero from the left?

2) What is the limit of the function as it approaches zero from the right?

3) What is the function value at zero and is it equal to the first two statements?

If we look at the graph we see that as  approaches zero from the left the  values approach zero as well. This is also true if we look the values as  approaches zero from the right. Lastly we look at the function value at zero which in this case is also zero.

Therefore, we can observe that  as  approaches .

Examining the graph above, we need to look at three things:

1) What is the limit of the function as  approaches zero from the left?

2) What is the limit of the function as  approaches zero from the right?

3) What is the function value as  and is it the same as the result from statement one and two?

Therefore, we can determine that  does not exist, since  approaches two different limits from either side :  from the left and  from the right. 

Examining the graph, we want to find where the graph tends to as it approaches zero from the right hand side. We can see that there appears to be a vertical asymptote at zero. As the x values approach zero from the right the function values of the graph tend towards positive infinity.

Therefore, we can observe that   as  approaches  from the right.

Examining the graph, we can observe that does not exist, as   is not continuous at . We can see this by checking the three conditions for which a function is continuous at a point :

1. A value exists in the domain of

2. The limit of exists as approaches

3. The limit of at is equal to

Given , we can see that condition #1 is not satisfied because the graph has a vertical asymptote instead of only one value for and is therefore an infinite discontinuity at .

We can also see that condition #2 is not satisfied because approaches two different limits:  from the left and from the right.

Based on the above, condition #3 is also not satisfied because is not equal to the multiple values of .

Thus, does not exist.