Question

consider the following function.

$f(x)=\frac{1}{x - 3}$

determine whether $f(x)$ approaches $infty$ or $-infty$ as $x$ approaches 3 from the left and from the right.
(a) $lim_{x
ightarrow3^{-}}f(x)$
(b) $lim_{x
ightarrow3^{+}}f(x)$

Explanation:

Step1: x→3⁻, x < 3, x-3 < 0

$f(x)=\frac{1}{\text{small negative}}$

Step2: 1/(small negative) → -∞

$\lim_{x→3^-} f(x) = -\infty$

Step3: x→3⁺, x > 3, x-3 > 0

$f(x)=\frac{1}{\text{small positive}}$

Step4: 1/(small positive) → ∞

$\lim_{x→3^+} f(x) = \infty$

Answer:

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