To evaluate the limit, we must first determine whether the limit is right or left sided; the plus sign "exponent" on  indicates that we are evaluating the limit from the right, or using values slightly larger than . The second function of the piecewise function corresponds to these values, and when we evaluate the limit using this function we get , as when the natural log function approaches zero, it approaches .

To evaluate the limit, we must first pull out a factor consisting of the highest power term divided by itself (so that we are leaving the limit unchanged):

Once the factor we pulled out divides to zero, and the negative exponent terms go to zero (as  approaches infinity, you get infinity in the denominator for each of those terms, which all approach zero), you are left with a constant, .

To evaluate the limit, we first must determine whether the limit is right or left sided; the negative sign "exponent" on the  indicates we are approaching  from the left, or with values slightly less than . The first function within the piecewise function corresponds to these values, and when we evaluate the limit, we get .

To evaluate the limit, we must first determine whether the limit is right or left sided; the plus sign "exponent" indicates that we are approaching values from the right side,  values slightly greater than . So, we must use the last function within the piecewise function, which corresponds to values of  greater than . When we evaluate the limit using this function, we get , because as the natural log function input gets closer to zero, the output approaches .

To evaluate the limit, we must first pull out a factor consisting of the highest power term divided by itself (so we are unchanging the contents of the limit):

Once the factor we pulled out becomes , and the negative exponent terms go to zero (as  approaches negative infinity, the entire term goes to zero), what we have left is a constant, .

To evaluate the limit, we must first pull out a factor consisting of the highest power term divided by itself (so we are unchanging the contents of the limit):

Once the factor we pulled out becomes , and all of the terms in the denominator go to zero (as  approaches infinity, the negative exponent terms become zero), we are left with .

In general, when you are looking for  you are looking to see whether the limit of  exists to the right, and if it does, what is the value.

Solution:

In this case, we want to see the limit at  , from the right. The limit exists, and the value corresponds with the function 

As you go to , the function will have a  for the denominator, which is not allowed.

Let's make sure the denominator is equal to if we were to plug in .

As we go to 4 from the right, the function is tending towards infinity.

We have positive infinity because when going to  from the right, the value will always be greater than  and so always positive.

Therefore, the answer is 

As you go to , the function will have a  for the denominator, which is not allowed.

To make sure, let's plug in  for the denominator.

As we go to  from the right, the function is tending towards infinity.

We have positive infinity because when going to  from the right, the value will always be greater than  and so always positive.

As you go to , the function will get closer and closer to having  for the denominator, sending the total value of the function into infinity. As we go to  from the left, the function is tending towards negative infinity. It is a negative because when going to  from the left, the function will always be less than , hence negative. 

Let's make sure the denominator is equal to zero if we were to plug in 

So we know that this limit will go to .