# Is #f(x)=(x^3+3x^2-4x-9)/(x+1)# increasing or decreasing at #x=0#?

graph{[-5, 5, -12, 5]}/(x+1) = (x^3 + 3x^2 - 4x - 9)

Repeat the preceding actions and add the current column:

It is now simpler to distinguish between these, so

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To determine whether ( f(x) = \frac{x^3 + 3x^2 - 4x - 9}{x + 1} ) is increasing or decreasing at ( x = 0 ), we can use the first derivative test. First, find the derivative of ( f(x) ), then evaluate the derivative at ( x = 0 ). If the derivative is positive at ( x = 0 ), the function is increasing at that point. If the derivative is negative, the function is decreasing.

( f'(x) = \frac{d}{dx}\left(\frac{x^3 + 3x^2 - 4x - 9}{x + 1}\right) )

Using the quotient rule:

( f'(x) = \frac{(x+1)(3x^2 + 6x - 4) - (x^3 + 3x^2 - 4x - 9)(1)}{(x + 1)^2} )

Simplify and evaluate ( f'(0) ):

( f'(0) = \frac{(0 + 1)(3(0)^2 + 6(0) - 4) - ((0)^3 + 3(0)^2 - 4(0) - 9)(1)}{(0 + 1)^2} )

( f'(0) = \frac{(1)(-4) - (-9)(1)}{(1)^2} )

( f'(0) = \frac{-4 + 9}{1} )

( f'(0) = \frac{5}{1} )

Since ( f'(0) = 5 > 0 ), the function is increasing 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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