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

graph{[0, 4.4, 5.808, 14.8, 15.2]} = -x-6-25/(x-5)-y)(24x-y-81)=0

By precise division,

At x = 4, y = -x-6-25/(x-5) = 15.

Thus, P(4, 15) is the tangent's point of contact.

At x = 4, the tangent's slope, m, is fi.

f#'=-1+25/(x-5)^2=24 =m.

Consequently, the tangent equation at P is

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To find the equation of the tangent line at x = 4, we need to find the slope of the tangent line and a point on the line.

First, let's find the slope. We can use the derivative of the function f(x) to find the slope at any given point.

The derivative of f(x) = 1 - x^2/(x - 5) can be found using the quotient rule. After simplifying, we get:

f'(x) = (2x^2 - 10x - 5)/(x - 5)^2

Now, let's find the slope at x = 4 by substituting x = 4 into the derivative:

f'(4) = (2(4)^2 - 10(4) - 5)/(4 - 5)^2 = (32 - 40 - 5)/(-1)^2 = -13

So, the slope of the tangent line at x = 4 is -13.

Next, let's find a point on the line. We can substitute x = 4 into the original function f(x):

f(4) = 1 - (4^2)/(4 - 5) = 1 - 16/(-1) = 1 + 16 = 17

Therefore, a point on the tangent line is (4, 17).

Finally, we can use the point-slope form of a line to find the equation of the tangent line:

y - y1 = m(x - x1)

Substituting the values we found, we get:

y - 17 = -13(x - 4)

Simplifying, we have:

y - 17 = -13x + 52

Rearranging the equation, we get the equation of the tangent line:

y = -13x + 69

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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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