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QuestionFind the limit as xx approaches -3 for the expression x2+9x+18x+3\frac{x^{2}+9x+18}{x+3}.

Studdy Solution

STEP 1

Assumptions1. We are given the function x+9x+18x+3\frac{x^{}+9 x+18}{x+3} . We are asked to find the limit as xx approaches −3-3

STEP 2

First, we try to substitute x=−x = - directly into the function.
lim⁡x→−x2+9x+18x+\lim{x \rightarrow-} \frac{x^{2}+9 x+18}{x+}

STEP 3

Substitute x=−3x = -3 into the function.
(−3)2+9(−3)+18−3+3\frac{(-3)^{2}+9 (-3)+18}{-3+3}

STEP 4

We see that the denominator becomes zero, which means the function is undefined at x=−3x = -3. This suggests that there may be a factor in the numerator that cancels out with the denominator.

STEP 5

To find this factor, we factorize the numerator.
x2+9x+18=(x+3)(x+)x^{2}+9 x+18 = (x+3)(x+)

STEP 6

Substitute the factored form of the numerator back into the function.
lim⁡x→−3(x+3)(x+6)x+3\lim{x \rightarrow-3} \frac{(x+3)(x+6)}{x+3}

STEP 7

Now, we can cancel out the common factor (x+3)(x+3) in the numerator and denominator.
lim⁡x→−3(x+6)\lim{x \rightarrow-3} (x+6)

STEP 8

Substitute x=−3x = -3 into the simplified function.
lim⁡x→−3(−3+6)\lim{x \rightarrow-3} (-3+6)

STEP 9

Calculate the limit.
lim⁡x→−3(−3+6)=3\lim{x \rightarrow-3} (-3+6) =3The limit of the function as xx approaches −3-3 is 33.

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