As x becomes infinitely large,  approaches .

The limit of a function as  approaches a value  exists if and only if the limit from the left is equal to the limit from the right; the actual value of  is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

, so .

For  to be continuous, it must hold that 

.

To find , we can use the definition of  for all negative values of :

It must hold that  as well; using the definition of  for all positive values of :

, so .

Now examine . For  to be differentiable, it must hold that 

.

To find , we can differentiate the expression for  for all negative values of :

To find , we can differentiate the expression for  for all positive values of :

We know that , so

Since  and ,

 cannot exist regardless of the values of  and . 

As x becomes infinitely large,  approaches infinity.

So for limits involving infinity, there is one important concept regarding fractions that is important to understand.

So if you have a fraction with a number on the bottom that is getting larger and larger, the whole fraction becomes smaller.

For example 

So the way to solve for limits involving infinity is to divide each term on the top, and each term on the bottom by the largest power of the variable. In the case of this problem, that is .

So if you do this you get:

So anything divided by itself is 1, which means that the first two terms cancel to one. Then the rest of the terms with a higher power of x on the bottom than on the top will have some power of x left on the bottom. This will look like:

Then, if you let the limit go to infinity, the terms with an x left on the bottom will go to zero. 

This leaves you with the answer of 

You can substitute  to write this as:

Note that as , 

, since the fraction becomes indeterminate, we need to take the derivative of both the top and bottom of the fraction.

, which is the correct choice.

Substitute  to rewrite this limit in terms of u instead of x. Multiply the top and bottom of the fraction by 2 in order to make this substitution:

(Note that as , .)

, so

, which is therefore the correct answer choice.

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution). 

To evaluate the limit, we must first determine whether we are approaching 4 from the left or right; the positive sign exponent indicates that this is a right side limit, or that we are approaching using values slightly greater than 4. When we substitute this into the function, we find that we approach , as the natural logarithm function approaches this when its domain goes to zero.