Question

find the limits in a) through c) below for the function f(x) = \frac{x^{2}-5x - 36}{x - 9}. use -\infty and \infty when appropriate.
a) select the correct choice below and fill in any answer boxes in your choice.
a. \lim_{x\to9^{-}}f(x)= (simplify your answer.)
b. the limit does not exist and is neither -\infty nor \infty.

Explanation:

Step1: Factor the numerator

Factor $x^{2}-5x - 36$ as $(x - 9)(x+4)$. So $f(x)=\frac{(x - 9)(x + 4)}{x - 9}$.

Step2: Simplify the function

Cancel out the common factor $(x - 9)$ for $x
eq9$. Then $f(x)=x + 4$ for $x
eq9$.

Step3: Find the left - hand limit

To find $\lim_{x
ightarrow9^{-}}f(x)$, substitute $x = 9$ into $y=x + 4$. Since $y=x + 4$ is a continuous function, $\lim_{x
ightarrow9^{-}}f(x)=\lim_{x
ightarrow9^{-}}(x + 4)=9+4 = 13$.

Answer:

A. $\lim_{x
ightarrow9^{-}}f(x)=13$