# What is the equation of the tangent line of #f(x) =x/(x-2e^x)+x/e^x-x# at #x=3#?

Not simplified *

Averaged to 4 decimal places

We identify that this function requires the use of the quotient rule multiple times to find the equation for the tangent line.

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To find the equation of the tangent line of the function f(x) = x/(x-2e^x) + x/e^x - x at x = 3, we need to find the derivative of the function and evaluate it at x = 3.

First, let's find the derivative of f(x).

f'(x) = (d/dx)[x/(x-2e^x)] + (d/dx)[x/e^x] - (d/dx)[x]

Using the quotient rule and the chain rule, we can simplify this expression:

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

Simplifying further:

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

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

Now, let's evaluate f'(x) at x = 3:

f'(3) = [(2(3)e^3 - e^3)/(3-2e^3)^2] + [(e^3 - 3e^3)/(e^3)^2] - 1

Simplifying further:

f'(3) = [(6e^3 - e^3)/(3-2e^3)^2] + [(-2e^3)/(e^6)] - 1

f'(3) = [(5e^3)/(3-2e^3)^2] - [2e^3/(e^6)] - 1

Now, we have the slope of the tangent line at x = 3. To find the equation of the tangent line, we use the point-slope form:

y - y1 = m(x - x1)

Using the point (3, f(3)) and the slope f'(3), we can substitute the values into the equation:

y - f(3) = f'(3)(x - 3)

Simplifying further:

y - [f(3)] = f'(3)(x - 3)

This is the equation of the tangent line of f(x) = x/(x-2e^x) + x/e^x - x at x = 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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