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

Dividing both sides by , we get:

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

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.

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.

In order for a function or graph to have a limit at a specific value the graph must be approach the same function value from each side of the specific value in question. For this particular question, we want to look at the graph as it approaches the x value zero from both the right and left hand side. Since the function values approach positive infinity from either side of zero we can conclude that the limit as x approaches zero is also positive infinity.

Thus by examining the graph, we can observe that   as  approaches  from the left and from the right.

In this particular problem we are asked to find a one sided limit of the function depicted in the graph. Since there is a plus sign on the exponent of the zero, this indicates that we are looking for a right hand limit. In other words, we will examine x values that are slightly larger than zero.

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

For this particular function we are only looking at x values that are slightly less than zero. This is indicated to us by the negative sign in the exponent on zero.

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

To find the limit at a particular point we need to find three things.

1. What is the function value at .

2. What function value does the graph approach as x approaches zero from the left?

3. What function value does the graph apprach as x approaches zero from the right?

If all three statements are equal to one another than the limit exists and is equal to the function value at the particular x value.

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

Evaluating this limit requires use of L'Hopital's Rule. But first, we must confirm the given problem statement is of indeterminate form:

If we let x approach infinity, the result turns into . This is an indeterminate form, and therefore, we can apply L'Hopital's Rule, by taking the derivative of the numerator and denominator separately.

If we now have x approach infinity, the result turns into 

This is one of the answer choices.

Evaluating this limit requires the use of L'Hopital's Rule, and a little bit of intuition to simplify the calculations. But first, let's confirm that L'Hopital's Rule can be used by checking if the limit is of indeterminate form. When x approaches infinity:

This is of indeterminate form, and therefore we can use L'Hopital's Rule.

Notice that the limit of this will also result in an indeterminate form of . Also notice that the bottom term will always remain the same no matter how many times we take the derivative, and will therefore always approach infinity, while the numerator will eventually reach the point where the 'x term' will disappear, and will at some point approach some constant:

If we keep taking the derivative of the top, eventually we'll get something of the form:

 where 

The product of these terms will be some constant, and we will therefore have a constant term in the numerator, and the denomator will always be approaching a bigger number. Therefore, the overall limit approaches:

This is one of the answer choices.