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 not equal zero when x=2; so we proceed to insert the value of x into the entire equation.

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, 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.

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

First, we can factor the numerator to obtain the following form.

The  term can now cancel on the numerator and denominator. Therefore, this problem becomes

Alternatively we can also use L'Hopital's rule since the limit of the following is not defined. L'Hopital's rule states to take the derivative of both the numerator and the denominator then substitute the value into the new fraction. Repeat those steps until the limit is found.

Taking the derivative of the numerator and denominator with respect to , we get

To evaluate the limit, we must first determine whether the limit is a right or left sided limit; the plus sign indicates that values slightly greater than 1 are being approached (from the right side), so the function that we use the second function, corresponding to values greater than or equal to 1. When we plug 1 into this function, we get our answer, .

To evaluate the limit, we must first determine whether the limit is being approached from the right or left. The negative sign "exponent" on 4 indicates that numbers slightly less than 4 are being approached, and that the limit is being evaluated from the left side. Now, simply use the first function (for values less than 4) to evaluate the limit. We see that we get  by subsituting 4 into the first function. 

To evaluate the limit, we must determine whether the limit is right or left sided. Because 0 has a negative sign "exponent", we know we are approaching from values slightly less than 0, or from the left side. Now, evaluate the limit using the part of the piecewise function corresponding to values less than (or equal to) 0, and you get .

The limiting situation in this equation would be the denominator. Plug the value that  is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=3; 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.

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=5; 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.