Question

find the limits in a), b), and c) below for the function f(x) = 1/(x - 8). use -∞ and ∞ when appropriate.
a) select the correct choice below and fill in any answer boxes in your choice.
a. lim f(x) = -∞
x→8⁻
(simplify your answer.)
b. the limit does not exist and is neither -∞ nor ∞.
b) select the correct choice below and fill in any answer boxes in your choice.
a. lim f(x) =
x→8⁺
(simplify your answer.)
b. the limit does not exist and is neither -∞ nor ∞.

Explanation:

Step1: Analyze left - hand limit

When $x\to8^{-}$, we consider values of $x$ that are less than 8. Let $x = 8 - h$, where $h>0$ and $h\to0$. Then $f(x)=\frac{1}{x - 8}=\frac{1}{(8 - h)-8}=-\frac{1}{h}$. As $h\to0$, $-\frac{1}{h}\to-\infty$.

Step2: Analyze right - hand limit

When $x\to8^{+}$, we consider values of $x$ that are greater than 8. Let $x = 8+h$, where $h > 0$ and $h\to0$. Then $f(x)=\frac{1}{x - 8}=\frac{1}{(8 + h)-8}=\frac{1}{h}$. As $h\to0$, $\frac{1}{h}\to\infty$.

Answer:

a) A. $\lim_{x\to8^{-}}f(x)=-\infty$
b) A. $\lim_{x\to8^{+}}f(x)=\infty$