# Is #f(x)=(-7x^3-x^2-2x+2)/(x^2+3x)# increasing or decreasing at #x=1#?

It is decreasing at

and applying the quotient formula to obtain

graph{(x^2+3x) [-4.58, 5.42, -4.1, 0.9]} / (-7x^3-x^2-2x+2)

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To determine if ( f(x) = \frac{-7x^3 - x^2 - 2x + 2}{x^2 + 3x} ) is increasing or decreasing at ( x = 1 ), we can evaluate the derivative of ( f(x) ) and check its sign at ( x = 1 ).

The derivative of ( f(x) ) with respect to ( x ) is given by: [ f'(x) = \frac{(2x + 3)(-7x^3 - x^2 - 2x + 2) - (x^2 + 3x)(-21x^2 - 2x - 2)}{(x^2 + 3x)^2} ]

Substitute ( x = 1 ) into ( f'(x) ) to find the sign: [ f'(1) = \frac{(2(1) + 3)(-7(1)^3 - (1)^2 - 2(1) + 2) - ((1)^2 + 3(1))(-21(1)^2 - 2(1) - 2)}{((1)^2 + 3(1))^2} ] [ f'(1) = \frac{(5)(-7 - 1 - 2 + 2) - (1 + 3)(-21 - 2 - 2)}{(1 + 3)^2} ] [ f'(1) = \frac{(5)(-8) - (4)(-25)}{16} ] [ f'(1) = \frac{-40 + 100}{16} ] [ f'(1) = \frac{60}{16} ] [ f'(1) = 3.75 ]

Since ( f'(1) = 3.75 > 0 ), the function ( f(x) ) is increasing 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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