Examining the graph of , we can see that as  approaches  from either side,  .

Examining the graph above, we can determine that  as  approaches  from the left.

The limit  indicates that we are trying to find the value of the limit as  approaches to zero from the right side of the graph.  

From right to left approaching , the limit approaches to 1 even though the value at  of the piecewise function does not exist.

The answer is .

By examining the above graph of , we can observe that as  approaches  from the left, 

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

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

2) What is the limit of the function as it approaches zero from the right?

3) What is the function value at zero and is it equal to the first two statements?

If we look at the graph we see that as  approaches zero from the left the  values approach zero as well. This is also true if we look the values as  approaches zero from the right. Lastly we look at the function value at zero which in this case is also zero.

Therefore, we can observe that  as  approaches .

Examining the graph of the function above, we can observe that there is a horizontal asymptote at . Now if we look at the function values as  approaches  we see that the  values tend to .

Therefore we can observe that  as  approaches .

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?

Looking at the graph we can determine that  as  approaches  because from both the left and right sides of zero, the function is approaching infinity.

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 determine that  does not exist, since  approaches two different limits from either side :  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?

We can observe that  does not exist, as  approaches two different limits:  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?

We can observe that  , as  approaches  from the left and from the right.