Question

consider the following function.
$f(x)=\frac{1}{x-8}$
determine whether $f(x)$ approaches $\infty$ or $-\infty$ as $x$ approaches 8 from the left and from the right.
(a) $\lim_{x \to 8^-} f(x)$
(b) $\lim_{x \to 8^+} f(x)$

Explanation:

Step1: Analyze left-hand limit

For $x \to 8^-$, $x$ is less than 8, so $x-8 < 0$. As $x$ gets closer to 8 from the left, $x-8$ approaches 0 from the negative side.
$\lim_{x \to 8^-} \frac{1}{x-8} = -\infty$

Step2: Analyze right-hand limit

For $x \to 8^+$, $x$ is greater than 8, so $x-8 > 0$. As $x$ gets closer to 8 from the right, $x-8$ approaches 0 from the positive side.
$\lim_{x \to 8^+} \frac{1}{x-8} = +\infty$

Answer:

(a) $-\infty$
(b) $+\infty$ (or $\infty$)