Evaluating the limit directly will produce an indeterminant solution of .

The limit definition of a derivative is . However, the alternative form, , better suits the given limit.

Let  and notice . It follows that .  

Thus, the limit is 

Evaluating the derivative directly will produce an indeterminant solution of .

The limit definition of a derivative is . However, the alternative form, , better suits the given limit.

Let  and notice . It follows that .  Thus, the limit is .

The correct answer is 11.

Taking the derivative of  involves the product rule, and the chain rule.

Substituting  into both sides of the derivative we get

.

If we recall the definition of a derivative of a function  at a point , one of the definitions is 

.

If we compare this definition to the limit 

we see that that this is the limit definition of a derivative, so we need to find the function  and the point  at which we are evaluating the derivative at. It is easy to see that the function is  and the point is . So finding the limit above is equivalent to finding .

We know that the derivative is , so we have

.

Write the definition of the limit.

Substitute .

Since  is approaching to zero, it would be best to evaluate when we assume that  is progressively decreasing.  Let's assume  and check the pattern.

The best answer is:  

Computation of the derivative requires the use of the Product Rule and Chain Rule. 

The Product Rule is used in a scenario when one has two differentiable functions multiplied by each other:

This can be easily stated in words as: "First times the derivative of the second, plus the second times the derivative of the first."

In the problem statement, we are given:

 is the "First" function, and  is the "Second" function. 

The "Second" function requires use of the Chain Rule. 

When:

Applying these formulas results in:

Simplifying the terms inside the brackets results in:

We notice that there is a common term that can be factored out in the sets of equations on either side of the "+" sign. Let's factor these out, and make the equation look "cleaner".

Inside the brackets, it is possible to clean up the terms into one expanded function. Let us do this:

Simplifying this results in one of the answer choices:

Recall that one definition for the derivative of a function  is .

This means that this question is asking us to find the value of the derivative of  at .

Since 

 and , the value of the limit is .

Evaluation of this integral requires use of the Product Rule. One must also need to recall the form of the derivative of .

Product Rule:

Applying these two rules results in:

This matches one of the answer choices.

In order to find , we need to remember how to find  by using the definition of derivative.

Definition of Derivative:

Now lets apply this to our problem.

Now lets expand the numerator.

We can simplify this to

Now factor out an h to get

We can simplify and then evaluate the limit.

In order to find , we need to remember how to find  by using the definition of derivative.

Definition of Derivative:

Now lets apply this to our problem.

Now lets expand the numerator.

We can simplify this to

Now factor out an h to get

We can simplify and then evaluate the limit.