# What is the equation of the line normal to # f(x)=2/(x-1)^2-2x+4# at # x=-2#?

Equation of normal is

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To find the equation of the line normal to the function f(x) at x=-2, we need to determine the slope of the tangent line at that point.

First, we find the derivative of f(x) using the power rule and chain rule:

f'(x) = -4(x-1)^(-3) - 2

Next, we substitute x=-2 into the derivative to find the slope of the tangent line at x=-2:

f'(-2) = -4(-2-1)^(-3) - 2

Simplifying this expression, we get:

f'(-2) = -4(-3)^(-3) - 2

Now, we can find the slope of the line normal to f(x) at x=-2 by taking the negative reciprocal of the slope of the tangent line:

m_normal = -1 / f'(-2)

Substituting the value of f'(-2) into the equation, we have:

m_normal = -1 / (-4(-3)^(-3) - 2)

Simplifying further, we get:

m_normal = -1 / (-4/27 - 2)

To simplify the denominator, we convert -2 into a fraction with a common denominator of 27:

m_normal = -1 / (-4/27 - 54/27)

Combining the fractions, we have:

m_normal = -1 / (-58/27)

To divide by a fraction, we multiply by its reciprocal:

m_normal = -1 * (27/-58)

Finally, we simplify the expression:

m_normal = 27/58

Therefore, the equation of the line normal to f(x) at x=-2 is y = (27/58)(x+2) + b, where b is the y-intercept of the line.

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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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