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

To evaluate the limit, we must first determine whether the limit is right or left sided. The negative sign "exponent" on 1 indicatest that we are approaching one using values slightly less than 1, or from the left. Thus, we must use the part of the piecewise function corresponding to x values less than (or equal to) 1. When we evaluate 

because the natural log function as it reaches zero equals negative infinity.

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 factor we pulled out cancels to become 1, and the negative power terms go to zero (infinity in the denominator makes zero), we are left with our final answer, .

First, we must determine whether the limit is being approached from the left or right. The negative sign "exponent" on the 10 indicates we are approaching 10 from the left, or using values slightly less than 10. Therefore we must use the piecewise function associated with values less than 10. When we evaluate the limit using this function, we get 10. 

Because we are given a piecewise function without a one-sided limit, it is tempting to declare "the limit does not exist," when in reality, when 4 is approached from the left or right, both functions have the same output at 4, which is .

We are given a piecewise function in which the limit differs depending on what side we approach 6 from. From the left side, we get a value of  but from the right we get . The limit does not exist because this conflict remains (we don't have a one-sided limit).

To evaluate the limit, we must see whether the limit is right or left sided. The plus sign "exponent" on the 0 indicates that we have a right sided limit, that we are approaching 0 using values slightly larger than 0. So, we must evaluate the limit using the part of the piecewise function corresponding to values greater than or equal to 0. When we do this, we get .

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,

 .