Limit Master

One-Sided & Infinite Limits

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One-Sided Limit Question #1

Evaluate the Limit:

Step-by-Step Explanation

Limit Rules & Infinite Limits Reference

1. One-Sided Limits & Existence

A two-sided limit exists if and only if both one-sided limits exist and are equal:

$\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = L \text{ and } \lim_{x \to a^+} f(x) = L$

2. Infinite Limits (Vertical Asymptotes)

When evaluating $\lim_{x \to a} \frac{k}{x - a}$ for $k \neq 0$:

$\frac{\text{Positive}}{\text{Small Positive } (0^+)} \to +\infty$
$\frac{\text{Positive}}{\text{Small Negative } (0^-)} \to -\infty$
If left-hand limit is $-\infty$ and right-hand limit is $+\infty$, two-sided limit is DNE.

3. Limits at Infinity ($x \to \pm\infty$)

For rational functions $f(x) = \frac{P(x)}{Q(x)}$:

$\text{deg}(P) < \text{deg}(Q)$: Limit is $0$.
$\text{deg}(P) = \text{deg}(Q)$: Ratio of leading coefficients.
$\text{deg}(P) > \text{deg}(Q)$: Grows to $+\infty$ or $-\infty$ (check signs).

4. Absolute Value Limit Trick

$|x - a| = \begin{cases} x - a & \text{if } x > a \\ -(x - a) & \text{if } x < a \end{cases}$

Hence $\lim_{x \to a^+} \frac{x-a}{|x-a|} = 1$ and $\lim_{x \to a^-} \frac{x-a}{|x-a|} = -1$.