find the limits in a) through c) below for the function f(x) = \\frac{x^{2}-6x - 36}{x - 9}. use -\\infty…

find the limits in a) through c) below for the function f(x) = \\frac{x^{2}-6x - 36}{x - 9}. use -\\infty and \\infty when appropriate.\na) select the correct choice below and fill in any answer boxes in your choice.\na. \\lim_{x\\to9^{-}}f(x)=13 (simplify your answer.)\nb. the limit does not exist and is neither -\\infty nor \\infty.\nb) select the correct choice below and fill in any answer boxes in your choice.\na. \\lim_{x\\to9^{+}}f(x)= (simplify your answer.)\nb. the limit does not exist and is neither -\\infty nor \\infty.

find the limits in a) through c) below for the function f(x) = \\frac{x^{2}-6x - 36}{x - 9}. use -\\infty and \\infty when appropriate.\na) select the correct choice below and fill in any answer boxes in your choice.\na. \\lim_{x\\to9^{-}}f(x)=13 (simplify your answer.)\nb. the limit does not exist and is neither -\\infty nor \\infty.\nb) select the correct choice below and fill in any answer boxes in your choice.\na. \\lim_{x\\to9^{+}}f(x)= (simplify your answer.)\nb. the limit does not exist and is neither -\\infty nor \\infty.

Answer

Explanation:

Step1: Factor the numerator

First, factor the numerator $x^{2}-5x - 36$. We know that $x^{2}-5x - 36=(x - 9)(x+4)$ since $(-9)\times4=-36$ and $-9 + 4=-5$. So the function $f(x)=\frac{x^{2}-5x - 36}{x - 9}=\frac{(x - 9)(x + 4)}{x - 9}$. For $x\neq9$, we can cancel out the $(x - 9)$ terms, and $f(x)=x + 4$.

Step2: Find the right - hand limit

To find $\lim_{x\rightarrow9^{+}}f(x)$, since $f(x)=x + 4$ for $x\neq9$, we substitute $x = 9$ into $x+4$. So $\lim_{x\rightarrow9^{+}}f(x)=\lim_{x\rightarrow9^{+}}(x + 4)=9+4=13$.

Answer:

a) A. $\lim_{x\rightarrow9^{-}}f(x)=13$ b) A. $\lim_{x\rightarrow9^{+}}f(x)=13$