Math  /  Calculus

QuestionFind the limit. limx9x2+5x36x+9\lim _{x \rightarrow-9} \frac{x^{2}+5 x-36}{x+9}
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. limx9x2+5x36x+9=5\lim _{x \rightarrow-9} \frac{x^{2}+5 x-36}{x+9}=5 (Type an integer or a simplified fraction.) B. The limit does not exist.

Studdy Solution

STEP 1

1. The limit in question involves a rational function, which may be simplified by factoring the numerator and canceling common factors with the denominator.
2. To evaluate the limit, we need to ensure that the denominator does not equal zero at the point of interest, or handle the zero appropriately if it does.
3. If the function simplifies such that the denominator is no longer zero at the point of interest, we can directly substitute the value.

STEP 2

1. Factor the numerator of the rational function.
2. Simplify the rational function by canceling common factors.
3. Evaluate the limit by direct substitution after simplification.

STEP 3

Factor the numerator x2+5x36x^2 + 5x - 36.
x2+5x36=(x+9)(x4) x^2 + 5x - 36 = (x + 9)(x - 4)

STEP 4

Rewrite the original limit expression using the factored form of the numerator.
limx9(x+9)(x4)x+9 \lim _{x \rightarrow -9} \frac{(x + 9)(x - 4)}{x + 9}

STEP 5

Cancel the common factor (x+9)(x + 9) in the numerator and the denominator.
limx9(x4) \lim _{x \rightarrow -9} (x - 4)

STEP 6

Evaluate the limit by direct substitution.
limx9(x4)=94=13 \lim _{x \rightarrow -9} (x - 4) = -9 - 4 = -13
Therefore, the limit is:
limx9x2+5x36x+9=13 \lim _{x \rightarrow -9} \frac{x^{2}+5 x-36}{x+9} = -13
The correct choice is:
A. limx9x2+5x36x+9=13 \lim _{x \rightarrow -9} \frac{x^{2}+5 x-36}{x+9} = -13

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