The derivative of  is .

Therefore the integral of

 where C is some constant.

The rule for taking the derivative of .

For this problem we need to remember to use the Chain Rule.

Since we are taking the derivative of,

  we need to take the derivative of the outside piece  keeping the inside piece the same , and then multiply the whole thing by the derivative of the inside piece .

Therefore the solution becomes:

,

.

The derivative of,

.

The derivative of 

Therefore, using the Chain Rule the derivative of the function will become the derivative of the outside piece keeping the original inside piece. Then multiplying that by the derivative of the inside piece.

  Recall the formal definition of the derivative:

When you evaluate this limit the output is f'(x). In this question f(x) = ln(x) so what this question is really saying is take the derivative of f(x) and evaluate it at 2.  

Take the derivative:

Substitute the value 2 into the derivative:

This limit is very simple(almost too simple) because it asks for the limit at a location where there is no discontinuity. Fortunately, this makes taking the limit trivial.

Substitute x=4 into the function to evaluate the limit.

As x becomes infinitely large, the x terms with the highest power dominate the function and the terms of lower order become negligible.  This means that near infinity, the x^5 term in the numerator and the 10X^5 term in the denominator are the only values necessary to evaluate the limit.

Simplify and evaluate:

First, factor x-2 out from the numerator and denominator.

At x=4 there is a discontinuity, so we must evaluate the limit from the right and left side to see if it exists.

Evaluated from the right:

From the left:

Because the limit from the right is not equal to the limit from the left, the function does not exist at x=4.

For this problem it's important to notice that e is raised to the power of negative x.  

It may be clearer to rewrite the function as:

As x grows large the function will approach zero.

Recall the definition of the derivative for a function f(x):

This limit returns the derivative of f(x). For the limit presented in the problem, f(x)=sinx and f(x+h)=sin(x+h), so it's really just asking us to take the derivative of sinx which is trivial.  

As the function approaches infinity, both the numerator and the denominator approach infinity.

This is an indeterminate form so we can apply L'Hospital's rule to take the limit.  After taking the derivative of the numerator and denominator we get the following limit:

Since the denominator approaches infinity and the numerator approaches zero, the limit of the function is zero as x approaches infinity.