# What is the equation of the line normal to # f(x)=e^(2x)/(e^x+4)# at # x=2#?

In point slope form it is

The slope of the tangent line is given by:

So the slope of the normal line is

Rewrite using algebra as you see fit. (Or as your grader demands.)

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To find the equation of the line normal to the function f(x) = e^(2x)/(e^x+4) at x = 2, we need to determine the slope of the tangent line at x = 2 and then find the negative reciprocal of that slope to obtain the slope of the normal line.

To find the slope of the tangent line, we can take the derivative of the function f(x) with respect to x and evaluate it at x = 2.

The derivative of f(x) = e^(2x)/(e^x+4) is given by:

f'(x) = (2e^(2x)(e^x+4) - e^(2x)(e^x))/(e^x+4)^2

Evaluating f'(x) at x = 2:

f'(2) = (2e^(4)(e^2+4) - e^(4)(e^2))/(e^2+4)^2

Now, we can find the slope of the tangent line by evaluating f'(2).

Next, we take the negative reciprocal of the slope of the tangent line to obtain the slope of the normal line.

Finally, using the point-slope form of a line, we can write the equation of the line normal to f(x) at x = 2.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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