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. 

First, try evaluating the limit at the target value.

This gives us an indeterminate form, so we have to keep trying. Let's factor the polynomials:

We can cancel an , so let's do that.

Now evaluate at the target value.

The limit evaluates to .

When the function is a constant to the power of a function of x, the first step in chain rule is to rewrite f(x).  So, the first factor of f(x) will be .  Next, we have to take the derivative of the function that is the exponent, or .  Its derivative is 10x-7, so that is the next factor of our derivative.  Last, when a constant is the base of an exponential function, we must always take the natural log of that number in our derivative.  So, our final factor will be .  Thus, the derivative of the entire function will be all these factors multiplied together: .

Whenever we have an exponential function with , the first term of our derivative will be that term repeated, without changing anything.  So, the first factor of the derivative will be .  Next, we use chain rule to take the derivative of the exponent.  Its derivative is .  So, the final answer is .

To take the derivative of any exponential function, we repeat the exponential function in the derivative.  So, the first factor of the derivative will be .  Next, we have to take the derivative of the exponent using chain rule.  The derivative of the trigonometric function secx is secxtanx, so in terms of this problem its derivative is .  Since the angle has a scalar of 3, we must also multiply the entire derivative by 3.  So, the answer is .