# What is the equation of the normal line of #f(x)=(x-1)^2/(x-5)# at #x=4 #?

By actual division,

y'=1-16/(x-5)^2=-15, at x = 4.

The slope of the normal = -1/y'=1/15.

graph{((x-1)^2/(x-5)-y)(x-15y-139)((x-4)^2+(y+9)^2-1)=0 [-80, 80, -40, 40]}

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To find the equation of the normal line of f(x)=(x-1)^2/(x-5) at x=4, we need to determine the slope of the tangent line at x=4 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 at x=4, we can use the derivative of the function f(x). Taking the derivative of f(x) with respect to x, we get:

f'(x) = (2(x-1)(x-5) - (x-1)^2) / (x-5)^2

Evaluating f'(x) at x=4, we have:

f'(4) = (2(4-1)(4-5) - (4-1)^2) / (4-5)^2

Simplifying this expression, we find:

f'(4) = -3

The slope of the tangent line at x=4 is -3.

To find the slope of the normal line, we take the negative reciprocal of -3, which is 1/3.

Now, we have the slope of the normal line, and we also know that it passes through the point (4, f(4)). To find f(4), we substitute x=4 into the original function:

f(4) = (4-1)^2 / (4-5)

Simplifying this expression, we get:

f(4) = 9

Therefore, the point (4, f(4)) is (4, 9).

Using the point-slope form of a linear equation, we can write the equation of the normal line:

y - 9 = (1/3)(x - 4)

Simplifying this equation, we obtain:

y = (1/3)x + 7/3

Hence, the equation of the normal line of f(x)=(x-1)^2/(x-5) at x=4 is y = (1/3)x + 7/3.

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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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