# Is #f(x)=(3x^3+3x^2+5x+2)/(x-2)# increasing or decreasing at #x=3#?

Decreasing; see explanation

Thus:

We can simplify further if we wish, but for the purposes of this problem it's unnecessary.

Viewing the graph below, we can verify this is correct.

graph{(3x^3 + 3x^2 + 5x + 2)/(x-2) [-2.947, 9.2, 121.787, 127.86]}

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To determine if ( f(x) = \frac{3x^3 + 3x^2 + 5x + 2}{x - 2} ) is increasing or decreasing at ( x = 3 ), we can use the first derivative test. If the derivative is positive at ( x = 3 ), then ( f(x) ) is increasing at that point. If the derivative is negative at ( x = 3 ), then ( f(x) ) is decreasing at that point.

The derivative of ( f(x) ) is ( f'(x) = \frac{d}{dx}\left(\frac{3x^3 + 3x^2 + 5x + 2}{x - 2}\right) ).

Using the quotient rule, we find:

( f'(x) = \frac{(x - 2)(9x^2 + 6x + 5) - (3x^3 + 3x^2 + 5x + 2)(1)}{(x - 2)^2} ).

Evaluating this at ( x = 3 ), we get:

( f'(3) = \frac{(3 - 2)(9(3)^2 + 6(3) + 5) - (3(3)^3 + 3(3)^2 + 5(3) + 2)}{(3 - 2)^2} ).

Simplifying:

( f'(3) = \frac{(1)(81 + 18 + 5) - (27 + 9 + 15 + 2)}{1} ),

( f'(3) = \frac{104 - 53}{1} ),

( f'(3) = \frac{51}{1} ),

( f'(3) = 51 ).

Since ( f'(3) = 51 ) is positive, ( f(x) ) is increasing 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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