# What is the equation of the line that is normal to #f(x)= (2-x)/sqrt( 2x+2) # at # x=2 #?

The given function

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To find the equation of the line that is normal to the function f(x) = (2-x)/sqrt(2x+2) 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 f(x) with respect to x and evaluate it at x=2.

The derivative of f(x) = (2-x)/sqrt(2x+2) is given by:

f'(x) = (2(sqrt(2x+2)) + (2-x)(1/2)(2x+2)^(-1/2)) / (2x+2)

Evaluating f'(x) at x=2:

f'(2) = (2(sqrt(2(2)+2)) + (2-2)(1/2)(2(2)+2)^(-1/2)) / (2(2)+2) = (2(sqrt(6)) + 0) / 6 = (2sqrt(6)) / 6 = sqrt(6) / 3

The slope of the tangent line at x=2 is sqrt(6) / 3.

To find the slope of the normal line, we take the negative reciprocal of the slope of the tangent line:

Slope of the normal line = -1 / (sqrt(6) / 3) = -3 / sqrt(6)

Now that we have the slope of the normal line, we can use the point-slope form of a line to find the equation. Since the line is normal to f(x) at x=2, we know that it passes through the point (2, f(2)).

Substituting x=2 into f(x), we get:

f(2) = (2-2)/sqrt(2(2)+2) = 0

Therefore, the point of intersection is (2, 0).

Using the point-slope form of a line, we have:

y - y1 = m(x - x1)

Substituting the values, we get:

y - 0 = (-3 / sqrt(6))(x - 2)

Simplifying, we obtain the equation of the line that is normal to f(x) at x=2:

y = (-3 / sqrt(6))(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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