# How do you find the equation of the tangent line to curve #y=(x-1)/(x+1)#, that are parallel to the line #x-2y=2#?

There are two such lines:

#y = 1/2x - 1/2#

#y = 1/2x + 7/2#

We start by noticing that the line can be converted to

#x - 2 = 2y -> y = 1/2x - 1#

This line has a slope of

We can find the derivative via the quotient rule.

#y' = (x + 1 - (x- 1))/(x +1)^2#

#y' = 2/(x+ 1)^2#

Set this to

#1/2 = 2/(x +1)^2#

#(x +1)^2 = 4#

#x + 1 = +- 2#

#x= 1 or -3#

The corresponding values of the function at these points are

#y(1) = (1- 1)/(1 + 1) = 0#

#y(-3) = (-3 - 1)/(-3 + 1) = 2#

Now we must use find the tangent line equations.

#y - 0 = 1/2(x - 1) -> y = 1/2x - 1/2#

#y - 2 = 1/2(x +3) -> y = 1/2x + 7/2#

Our answer is clearly viable which can be proved graphically.

Hopefully this helps!

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To find the equation of the tangent line to the curve y=(x-1)/(x+1) that is parallel to the line x-2y=2, we need to find the derivative of the curve and then use it to determine the slope of the tangent line.

The derivative of y=(x-1)/(x+1) is given by the quotient rule:

dy/dx = [(x+1)(1) - (x-1)(1)] / (x+1)^2

Simplifying this expression, we get:

dy/dx = 2 / (x+1)^2

Since the tangent line is parallel to the line x-2y=2, it will have the same slope. The slope of the line x-2y=2 can be determined by rearranging the equation in slope-intercept form:

-2y = -x + 2 y = (1/2)x - 1

Comparing this equation to y=(x-1)/(x+1), we can see that the slope of the line is 1/2.

Therefore, the equation of the tangent line to the curve y=(x-1)/(x+1) that is parallel to the line x-2y=2 is:

y = (1/2)x + b

To find the value of b, we substitute the coordinates of a point on the curve into the equation. Let's choose the point (2, 1):

1 = (1/2)(2) + b 1 = 1 + b b = 0

Thus, the equation of the tangent line is:

y = (1/2)x + 0 y = (1/2)x

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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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