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.

Letand 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.

Letand notice. It follows that. Thus, the limit is.

The correct answer is 11.

Taking the derivative ofinvolves the product rule, and the chain rule.

Substitutinginto both sides of the derivative we get

.

If we recall the definition of a derivative of a functionat 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 functionand the point我们正在评估的导数。我s easy to see that the function isand 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.

Sinceis approaching to zero, it would be best to evaluate when we assume thatis progressively decreasing. Let's assumeand check the pattern.

The best answer is:

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

产品的规则中使用一个场景,一个哈s 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, andis 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 functionis.

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

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 findby 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 findby 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.