Because we are given a regular limit and a piecewise function, in order for the limit to exist, all of the functions must result in the same value when the x value is being approached (from all sides). In the given piecewise function, when x is less than 5, and equal to 5, when the limit is evaluated we get 5 as our answer. However, when x is greater than 5, we get -5 as our answer. Because of this ambiguity, the limit does not exist. 

Examining the graph, we can observe that does not exist, as   is not continuous at . We can see this by checking the three conditions for which a function is continuous at a point :

1. A value exists in the domain of

2. The limit of exists as approaches

3. The limit of at is equal to

Given , we can see that condition #1 is not satisfied because the graph has a vertical asymptote instead of only one value for and is therefore an infinite discontinuity at .

We can also see that condition #2 is not satisfied because approaches two different limits:   from the left and  from the right.

Based on the above, condition #3 is also not satisfied because is not equal to the multiple values of .

Thus, does not exist.

To evaluate the limit, we must first see whether it is right or left sided. The plus sign "exponent" on 2 means the limit is right sided, or that we are approaching 2 using values slightly greater than 2. So, we evaluate the limit using the part of piecewise function corresponding to values greater than or equal to 2, and when we substitute, we get our answer,

 .

To evaluate the limit, we must first pull out a factor consisting of the highest term divided by itself (so we are pulling out a 1):

After the factor we pulled out cancels, and the negative exponent term in the denominator goes to zero as x approaches infinity, we are left with our answer, .

To evaluate the limit, we must first pull out a factor consisting of the highest power term divided by the highest power term (so we are pulling out a factor of 1):

After the term we pulled out cancels to 1, we can see that all of the negative power terms go to zero as the x approaches infinity (infinity in the denominator makes zero), so we are left with .

Examining the graph, we can observe that does not exist, as   is not continuous at . We can see this by checking the three conditions for which a function is continuous at a point :

1. A value exists in the domain of

2. The limit of exists as approaches

3. The limit of at is equal to

Given , we can see that condition #1 is not satisfied because the graph has a vertical asymptote instead of only one value for and is therefore an infinite discontinuity at .

We can also see that condition #2 is not satisfied because approaches two different limits:  from the left and from the right.

Based on the above, condition #3 is also not satisfied because is not equal to the multiple values of .

Thus, does not exist.

Examining the graph, we can observe that  as  approaches  from the left and from the right.

We can see this by checking the three conditions for which a function  is continuous at a point : 

1. A value  exists in the domain of 

2. The limit of  exists as  approaches 

3. The limit of  at  is equal to   

Since our , , and the limit as x approaches c from either side is also  we can conclude that .

This question is asking us to examine the graph from one side. The plus sign in the exponent on zero tells us that we want to look at the function values for x values that are slightly larger than zero.

Examining the graph, we can observe that  as  approaches  from the right.

This question is asking use to examine the graph from one side. The negative sign in the exponent on zero means we should look at the function values for x values that are slightly less than zero.

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

This question is asking us to examine the graph from one side. Since there is a plus sign in the exponent on the zero we want to look at the function values associated with x values that are slightly larger than zero.

Examining the graph, we can observe that  as  approaches  from the right.