Question

f(x) = \frac{1}{x - 3}
determine whether f(x) approaches \infty or -\infty as x approaches 3 from the left and from the right.
(a) \lim_{x \to 3^-} f(x)
\boxed{}
(b) \lim_{x \to 3^+} f(x)
\boxed{}

Explanation:

Step1: Analyze left-hand limit

For $x \to 3^-$, $x < 3$, so $x-3 < 0$. As $x$ approaches 3 from the left, $x-3$ approaches 0 from the negative side.
$\lim_{x \to 3^-} \frac{1}{x-3} = -\infty$

Step2: Analyze right-hand limit

For $x \to 3^+$, $x > 3$, so $x-3 > 0$. As $x$ approaches 3 from the right, $x-3$ approaches 0 from the positive side.
$\lim_{x \to 3^+} \frac{1}{x-3} = +\infty$

Answer:

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