This question is asking us to examine the graph from one side. Since there is a negative sign in the exponent on the zero we want to look at the function values associated with x values that are slightly less than zero.

Examining the graph, we can observe that  as  approaches  from the left. 

Since we have

we can plug this into the limit above to get

.

We cannot just plug in  into the function because then we get .

But we can do the following algebraic manipulation:

It is obvious that this function has a horizontal asymptote on both sides, but this can be seen using algebraic manipulation:

because the  approaches  as .

We can evaluate this limit using algebraic manipulation:

The denominator approaches  and the numerator approaches , so overall the limit approaches :

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.

For a limit to exist at a particular x value, the limit from the left of that value and the limit from the right of that value must be the same. Also the function value at that particular value must exist and be equal to the right and left sided limits.

Examining the graph, we can observe that   as  approaches  from the left and from the right.

First we need to identify which limit this question is asking us for. Since ther is a plus sign in the exponent on the zero, this means we are being asked to find the right hand limit of the function as it approaches zero. To find this, we will look at x values that are slightly larger than zero.

Examining the graph, we can observe that   as  approaches  from the right.

To evaluate the limit, we must first see if the limit is right or left sided. The negative sign "exponent" on  indicates we are approaching it from the left, or approaching values slightly less than . Accordingly, we must use the part of the piecewise function corresponding to x values less than (or equal to) . When we evaluate the limit using the correct function (the first one), we get .

We can convert from rectangular to polar form by using the following trigonometric identities: and . Given , then: