# Is #f(x)=(1-x)/(x+2)^3# increasing or decreasing at #x=-1#?

The function is increasing at

Since this is a positive, the function is growing.

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To determine if the function ( f(x) = \frac{1-x}{(x+2)^3} ) is increasing or decreasing at ( x = -1 ), we need to analyze the sign of its derivative at that point. We can find the derivative of ( f(x) ) with respect to ( x ) and then evaluate it at ( x = -1 ).

The derivative of ( f(x) ) is:

[ f'(x) = \frac{d}{dx} \left( \frac{1-x}{(x+2)^3} \right) ]

To find ( f'(x) ), we can use the quotient rule:

[ f'(x) = \frac{(x+2)^3 \cdot (-1) - (1-x) \cdot 3(x+2)^2}{(x+2)^6} ]

Now, we substitute ( x = -1 ) into ( f'(x) ):

[ f'(-1) = \frac{(-1+2)^3 \cdot (-1) - (1-(-1)) \cdot 3(-1+2)^2}{(-1+2)^6} ]

[ f'(-1) = \frac{(1)^3 \cdot (-1) - (2) \cdot 3(1)^2}{(1)^6} ]

[ f'(-1) = \frac{-1 - 6}{1} ]

[ f'(-1) = -7 ]

Since the derivative ( f'(-1) = -7 ) is negative, the function ( f(x) = \frac{1-x}{(x+2)^3} ) is decreasing at ( x = -1 ).

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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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