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

Answer 1

Equation of tangent is #9y+(e^2-4)x=5e^2-11#

The slope of the tangent is the same as the slope of curve at that point, which is given by the value of derivative of the function at that point.

Hence, slope of tangent to curve #f(x)=(e^x-1)/(x^2-1)# at #x=2# is given by #f'(2)#. And as the tangent passes through #(x,f(x))# i.e. #(2,f(2))#, the equation of tangent is
#y-f(2)=f'(2)(x-2)#
Now #f(2)=(e^2-1)/(2^2-1)=(e^2-1)/3#
and using quotient rule #f'(x)=((x^2-1)e^x-(e^x-1)2x)/(x^2-1)^2#
= #(x^2e^x-e^x-2xe^x+2x)/(x^2-1)^2#
and slope at #f(2)# is #f'(2)=(4e^2-e^2-4e^2+4)/9=(4-e^2)/9#

Hence, equation of tangent is

#y-(e^2-1)/3=(4-e^2)/9(x-2)#
or #9y-3e^2+3=4x-8-e^2x+2e ^2#
or #9y+(e^2-4)x=5e^2-11#

graph{(9y+(e^2-4)x-5e^2+11)(x^2y-y-e^x+1)=0 [-4.785, 5.215, -1.18, 3.82]}

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Answer 2

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

First, we find the derivative of f(x) using the quotient rule:

f'(x) = [(e^x)(x^2-1) - (e^x)(2x)] / (x^2-1)^2

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

f'(2) = [(e^2)(2^2-1) - (e^2)(2(2))] / (2^2-1)^2

Finally, we have the slope of the tangent line at x=2. To find a point on the line, we substitute x=2 into f(x):

f(2) = (e^2-1)/(2^2-1)

Therefore, the equation of the tangent line of f(x) = (e^(x)-1)/(x^2-1) at x=2 is:

y - f(2) = f'(2)(x - 2)

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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.

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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