Examining the graph above, we need to look at three things:

1) What is the limit of the function as  approaches three from the left?

2) What is the limit of the function as  approaches three 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 as  approaches :   from the left and  from the right.

Note that if you plug in  to the original limit, you get  as your answer. This shows that you need to use L'Hopital's rule and take the derivative of both the top and the bottom of the limit and then attempt to retry finding the limit.

The derivative of the top side becomes  and the derivative of the bottom side becomes  (the 1's go away because they are constants) and so you can rewrite the problem as so: 

.

Note that 1 raised to any power is just 1, so the limit becomes 

 which is . 

Examining the graph above, we need to look at three things:

1) What is the limit of the function as  approaches two from the left?

2) What is the limit of the function as  approaches two 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 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.