Computing this limit will require use of L'Hopital's Rule. To confirm that this rule can be used we must first plug in 0 into the given problem statement to see if it will result in an indeterminate form:

Because the result is the form of "0/0", we are allowed to use L'Hopital's Rule. We simply take the derivative of the numerator and the denominator separately like so:

Now, if we plug in 0 to the x terms we result in:

This is one of the answer choices.

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

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

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?

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

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 observe that  does not exist, as  approaches two different limits:  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 left hand side. We can see that there appears to be a vertical asymptote at zero. As the x values approach zero from the left, the function values of the graph tend towards positive infinity.

Thus, we can observe that  as  approaches  from the left.

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 observe that  as  approaches  from the left and from the right.

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 observe that does not exist, as  approaches two different limits:  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 left hand side. We can see that there appears to be a vertical asymptote at zero. As the x values approach zero from the left, the function values of the graph tend towards negative infinity.

Thus, we can observe that  as  approaches  from the left.

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?

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