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 it approaches two different limits:  from the left and   from the right.

For this problem we want to find the y values for which the graph tends towards as x approaches infinity. We can see that as the x values get larger the y values approach negative infinity quickly.

Therefore, by examining the graph we can observe that    as  approaches .

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?

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.

To evaluate the limit, we must first see whether the number 5 is being approached from the right or left. The minus sign indicates the limit is on the left side, meaning numbers less than 5 are being approached until 5 is reached. The function, therefore, that we need to use is the first one in the piecewise function:

Simply substitute in 5 into this function to get an answer of 

This limit can be approached using the techniques learned in Calculus I, or by applying the new concepts learning in Calculus II using L'Hopital's Rule. Both ways will result in the same answer. 

Using the Calculus I methodology, we can approach this problem like so:

Because we have an indeterminate form of "0/0", we can do some factoring and cancelling like so:

Therefore, the final result is 6, which agrees with one of the answer choices.

Using the Calculus II methodology, we can approach this problem like so:

Because we know that plugging in 3 directly into the original problem statement will give an indeterminate form, we can apply L'Hopital's Rule directly. 

First we differentiate the numerator and denominator separately:

Now, if we plug in 3 directly into this new statement, we will arrive at:

We see that we arrive at the same answer. 

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.