# Is #f(x)=(x^2+2x-2)/(2x-4)# increasing or decreasing at #x=0#?

The function is decreasing.

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To determine whether ( f(x) = \frac{x^2 + 2x - 2}{2x - 4} ) is increasing or decreasing at ( x = 0 ), we need to examine the sign of the derivative of ( f(x) ) at ( x = 0 ). If the derivative is positive, the function is increasing at that point; if negative, it's decreasing.

To find the derivative, ( f'(x) ), we use the quotient rule: [ f'(x) = \frac{(2x - 4)(2x + 2) - (x^2 + 2x - 2)(2)}{(2x - 4)^2} ]

Simplify the expression: [ f'(x) = \frac{4x^2 - 8 + 4x^2 + 4x - 4x^2 - 4x + 4}{(2x - 4)^2} ] [ f'(x) = \frac{8x - 4}{(2x - 4)^2} ]

Now, substitute ( x = 0 ) into ( f'(x) ) to find the sign of the derivative at ( x = 0 ): [ f'(0) = \frac{8(0) - 4}{(2(0) - 4)^2} ] [ f'(0) = \frac{-4}{(-4)^2} ] [ f'(0) = \frac{-4}{16} ] [ f'(0) = -\frac{1}{4} ]

Since ( f'(0) ) is negative, ( f(x) ) is decreasing at ( x = 0 ).

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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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