In general, when you are looking for  you are looking to see whether the limit of y 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 to the function 

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

Solution:

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

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

Solution:

In this case, we want to see the limit at , from the left. The limit exists, and the value correponds to the function 

To evaluate the limit, we must first determine whether the limit is right or left sided. The positive sign "exponent" on 7 indicates that we are evaluating the limit from the right side, or using numbers slightly larger than 7. The part of the piecewise function corresponding to these values is the second function; when we evaluate the limit using that function, we approach  (the natural log function approaches negative infinity as x approaches zero).

The limit exists if 

because 

we see that 

because 

we see that 

since 

we conclude that the limit exists and 

To evaluate the limit, we must make sure that the same value is being approached from both sides. When we evaluate the limit from the left (the first part of the piecewise function) and the right (the second part of the piecewise function), we get the same value () so the limit is equal to .

To evaluate the limit, we must first determine whether the limit is right or left sided; the positive sign "exponent" on the  indicates that the limit is right sided, or that we are approaching  with values slightly greater than . This corresponds to the second half of the piecewise function, and when we evaluate the limit using that function we get . 

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 .