The expression

means the limit of the function as x approaches 3 from the right.

Because

we see that 

There is no limiting situation in this equation (like a denominator) so we can just plug in the value that x approaches into the limit and solve:

There is no limiting situation in this equation (like a denominator) so we can just plug in the value that x approaches into the limit and solve:

There is no limiting situation in this equation (like a denominator) so we can just plug in the value that x approaches into the limit and solve:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will not equal zero when x=4; so we proceed to insert the value of x into the entire equation.

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 determine whether it is right or left sided. The negative sign "exponent" on the 2 indicates that we are approaching from the left, or numbers slightly less than 2. So, we choose the part of the piecewise function that is for x values less than or equal to 2. Now, evaluating the limit, as natural log approaches zero, we get . 

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

After the factor we pulled out cancels to 1, and all of the negative exponent terms go to zero (as they approach infinity), what is left is our answer .

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

Once the term we pulled out becomes zero, we then look at the remaining terms. The denominator of the fraction goes to zero because the sum of the two terms with negative exponents is .

Thus, the entire limit goes to .

To evaluate the limit, we must first see whether it is right or left sided.

The negative sign "exponent" on the 1 indicates we are approaching one from the left side, or with numbers slightly less than 1.

Therefore, we must use the part of the piecewise function corresponding to numbers less than (or equal to) 1. When we evaluate the limit, we find that

 because the exponent of e is infinity.