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 negative infinity.

Thus, we can observe that  as  approaches   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?

Using these rules 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?

Using these rules, we can observe that does not exist, as  approaches two different limits:  from the left and  from the right.

Evaluating this limit requires use of L'Hopital's Rule. However, in order to ensure this rule can be used, you must first make sure it is of  or  form. 

This is of indeterminate form and so, we may now use L'Hopital's Rule, by taking the derivative of the numerator and denominator separately:

Again, we arrive at an indeterminate form of , so we need to use L'Hopital's Rule once again:

Again, we arrive at an indeterminate form of , so we need to use L'Hopital's Rule once again:

This is one of the answer choices.

To evaluate the limit, first we must determine if it is right or left sided; the plus sign "exponent" on the 5 indicates that it is right sided, or that we are approaching values slightly greater than 5. Now, we use the corresponding function in the piecewise function to evaluate the limit, the function being the one for x values greater than or equal to 5. When we evaluate the limit by substitution, 

To evaluate the limit, we must determine which part of the piecewise function corresponds to the number we are approaching. Since x approaches positive infinity, we must use the function corresponding to x values greater than zero; when we directly substitute, we get . 

To evaluate the limit, we first must determine whether the limit is right or left sided. The negative sign "exponent" on the 3 indicates that we are approaching 3 from the left, or approaching numbers slightly less than 3. Therefore, we must evaluate the limit using the part of the piecewise function corresponding to numbers less than or equal to 3.

When we try to evaluate the limit using normal methods, we get the indeterminate form . So, we must use L'Hopital's Rule to solve the limit, which states that

for g(a) and h(a) both equal to zero. 

Now, find the derivatives of the numerator and denominator and evaluate the limit:

The derivatives were found using the following rules:

, 

There is no limiting situation in this equation (like a denominator) so we can just plug in the value that x approaches into the limit and solve:

There is no limiting situation in this equation (like a denominator) so we can just plug in the value that x approaches into the limit and solve:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=0; so we try to eliminate the denominator by factoring. When the denominator is no longer zero, we may continue to insert the value of x into the remaining equation.