One-Sided Limits

By the end of this lesson, you will be able to compute one-sided limits rigorously, use them to determine the two-sided limit, and analyze piecewise and absolute-value functions.

Definition

$$\lim_{x \to c^-}f(x) = L$$: the limit as $x$ approaches $c$ from the left.

$$\lim_{x \to c^+}f(x) = L$$: the limit as $x$ approaches $c$ from the right.

Theorem: $\lim_{x \to c}f(x) = L$ iff both $\lim_{x \to c^-}f(x) = L$ and $\lim_{x \to c^+}f(x) = L$.

Example 1 — Piecewise Function

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

$\lim_{x \to 2^-}f(x) = 4+1 = 5$; $\lim_{x \to 2^+}f(x) = 6-1 = 5$

Both equal 5 → $\lim_{x \to 2}f(x) = 5$ ✓ (even though this is checking continuity too)

Example 2 — Absolute Value

$\lim_{x \to 3}\dfrac{|x-3|}{x-3}$

$x > 3$: $|x-3| = x-3$, so $f(x) = 1 \to \lim_{x \to 3^+} = 1$

$x < 3$: $|x-3| = -(x-3)$, so $f(x) = -1 \to \lim_{x \to 3^-} = -1$

$1 \neq -1$ → limit does not exist.

Example 3 — Radical Function

$\lim_{x \to 0^+}\dfrac{x}{\sqrt{x}} = \lim_{x \to 0^+}\sqrt{x} = 0$

Only the right-hand limit exists (the function is undefined for $x < 0$).

Infinite One-Sided Limits and Vertical Asymptotes

If $\lim_{x \to c^+}f(x) = +\infty$ and $\lim_{x \to c^-}f(x) = +\infty$: we write $\lim_{x \to c}f(x) = +\infty$ (informal, meaning it's unbounded).

$\dfrac{1}{x}$ near $x = 0$:

$\lim_{x \to 0^+}1/x = +\infty$; $\lim_{x \to 0^-}1/x = -\infty$

These differ → the two-sided limit DNE. Vertical asymptote at $x = 0$.

Summary

One-sided limits: approach $c$ from one side only.

Two-sided limit: exists iff both one-sided limits agree.

Absolute value: write as piecewise; left and right limits often differ.

Infinite one-sided limits: indicate vertical asymptotes (not equal to the two-sided limit unless both sides agree on $\pm\infty$).