Question

find the limits in a) through c) below for the function $f(x)=\frac{x^{2}-11x + 10}{x + 10}$. use $-infty$ and $infty$ when appropriate.
a) select the correct choice below and fill in any answer boxes in your choice.
a. $lim_{x
ightarrow - 10^{-}}f(x)=-infty$ (simplify your answer.)
b. the limit does not exist and is neither $-infty$ nor $infty$.
b) select the correct choice below and fill in any answer boxes in your choice.
a. $lim_{x
ightarrow - 10^{+}}f(x)=square$ (simplify your answer.)
b. the limit does not exist and is neither $-infty$ nor $infty$.

Explanation:

Step1: Factor the numerator

First, factor \(x^{2}-11x + 10=(x - 10)(x - 1)\). So \(f(x)=\frac{(x - 10)(x - 1)}{x + 10}\).

Step2: Analyze the left - hand limit as \(x\to - 10^{-}\)

As \(x\to - 10^{-}\), the numerator \((x - 10)(x - 1)\to(-10 - 10)(-10 - 1)=(-20)\times(-11) = 220>0\), and the denominator \(x + 10\to0^{-}\). So \(\lim_{x\to - 10^{-}}f(x)=-\infty\).

Step3: Analyze the right - hand limit as \(x\to - 10^{+}\)

As \(x\to - 10^{+}\), the numerator \((x - 10)(x - 1)\to(-10 - 10)(-10 - 1)=220>0\), and the denominator \(x + 10\to0^{+}\). So \(\lim_{x\to - 10^{+}}f(x)=\infty\).

Answer:

a) A. \(\lim_{x\to - 10^{-}}f(x)=-\infty\)
b) A. \(\lim_{x\to - 10^{+}}f(x)=\infty\)