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\to - 10^{-}}f(x)=\square
(simplify your answer.)
b. the limit does not exist and is neither -\infty nor \infty.

Explanation:

Step1: Factor the numerator

Factor $x^{2}-11x + 10$ as $(x - 10)(x - 1)$. So $f(x)=\frac{(x - 10)(x - 1)}{x + 10}$.

Step2: Analyze the left - hand limit

As $x\to - 10^{-}$, we consider values of $x$ that are slightly less than $-10$. The numerator $(x - 10)(x - 1)$ approaches $(-10-10)(-10 - 1)=(-20)\times(-11)=220$. The denominator $x + 10$ approaches a small negative number.

Step3: Calculate the limit

Since the numerator is a non - zero constant and the denominator approaches $0$ from the negative side, $\lim_{x\to - 10^{-}}\frac{(x - 10)(x - 1)}{x + 10}=-\infty$.

Answer:

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