One-Sided Limits and Limits at Infinity

By the end of this lesson, you will be able to evaluate one-sided limits, connect them to the two-sided limit, evaluate limits at infinity for polynomials and rational functions, and identify horizontal asymptotes as limits.

One-Sided Limits

Left-hand limit: $\lim_{x \to c^-} f(x) = L$ — approach $c$ from below.

Right-hand limit: $\lim_{x \to c^+} f(x) = L$ — approach $c$ from above.

Key theorem: $\lim_{x \to c} f(x) = L$ iff both one-sided limits exist and equal $L$.

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

Example 1 — One-Sided Limits of a Piecewise Function

$$f(x) = \begin{cases} 2x + 1 & x < 3 \ x^2 - 4 & x \geq 3 \end{cases}$$

$\lim_{x \to 3^-} f(x) = 2(3) + 1 = 7$

$\lim_{x \to 3^+} f(x) = 9 - 4 = 5$

$7 \neq 5$ → $\lim_{x \to 3} f(x)$ does not exist.

Example 2 — Finding When the Two-Sided Limit Exists

$f(x) = \begin{cases} 3x - k & x \leq 2 \ x^2 + 1 & x > 2 \end{cases}$. Find $k$ so that $\lim_{x \to 2}$ exists.

Left: $3(2) - k = 6 - k$; Right: $4 + 1 = 5$

$6 - k = 5 \implies k = 1$

Infinite Limits

When $f(x) \to \pm\infty$ as $x \to c$:

$$\lim_{x \to c^+} \frac{1}{x-c} = +\infty, \quad \lim_{x \to c^-} \frac{1}{x-c} = -\infty$$

These indicate vertical asymptotes at $x = c$.

Note: Infinite limits technically don't "exist" in the real number system — but we write them as $\pm\infty$ to indicate the direction of growth.

Limits at Infinity

As $x \to +\infty$ or $x \to -\infty$:

For polynomials: the leading term dominates.

$$\lim_{x \to \infty} (3x^4 - 7x^2 + 1) = \lim_{x \to \infty} 3x^4 = +\infty$$

For rational functions: compare degrees of numerator and denominator.

Degree comparison	$\lim_{x \to \pm\infty} f(x)$
Num $<$ Den	$0$
Num $=$ Den	Ratio of leading coefficients
Num $>$ Den