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How do you find an equation of the tangent line to the graph of #f(x) = 1/(x-1)# at the point (2,1)?

Answer 1

Emily Ammerman

Since #f(x)=(x-1)^{-1}#, #f'(x)=-(x-1)^{-2}=-1/((x-1)^2)# so that #f'(2)=-1#.

Since #f(2)=1#, the equation of the tangent line is #y=f'(2)(x-2)+f(2)=-(x-2)+1=-x+3#

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Answer 2

Isaiah Allen

To find the equation of the tangent line to the graph of f(x) = 1/(x-1) at the point (2,1), we need to find the slope of the tangent line and the coordinates of the point.

First, we find the derivative of f(x) using the quotient rule: f'(x) = -1/(x-1)^2.

Next, we substitute x = 2 into f'(x) to find the slope of the tangent line at x = 2: f'(2) = -1/(2-1)^2 = -1.

Now, we have the slope of the tangent line, which is -1, and the point (2,1). We can use the point-slope form of a linear equation to find the equation of the tangent line:

y - y1 = m(x - x1), where (x1, y1) is the point (2,1) and m is the slope -1.

Substituting the values, we get: y - 1 = -1(x - 2).

Simplifying, we have: y - 1 = -x + 2.

Rearranging the equation, we get the equation of the tangent line: y = -x + 3.

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Answer from HIX Tutor

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.