One-Sided Limits

The central idea

The right-hand limit $\lim_{x\to a^+}f(x)=L$ uses inputs $x>a$, while the left-hand limit $\lim_{x\to a^-}f(x)=L$ uses $x<a$. A finite two-sided limit exists exactly when both one-sided limits exist and are equal. Neither one-sided limit depends on the isolated value $f(a)$.

Definitions, hypotheses, and notation

The superscript belongs to the input, not the output: $x\to a^-$ means values smaller than $a$, regardless of whether the resulting function values are positive or negative. This is especially important for reciprocal factors, where the side determines the denominator's sign and therefore whether the output tends to $+\infty$ or $-\infty$.

At an endpoint of a restricted domain, a one-sided limit may be the only meaningful local question. Continuity on $[a,b]$ uses the right-hand limit at $a$ and left-hand limit at $b$. A two-sided limit statement implicitly requires domain points approaching from both sides; when one side is absent, it should not be fabricated.

Conceptual meaning

One-sided limits describe how a graph approaches a boundary, jump, or piecewise join. They are also the correct language for endpoints of domains: $\sqrt x$ has a right-hand limit at zero even though no real-domain left side is available.

A dependable method and decision rule

1. Mark which side the superscript requests.
2. Use only the formula valid on that side of a piecewise definition.
3. Approach with sample inputs that stay inside the domain.
4. Evaluate or simplify the appropriate expression.
5. For a two-sided limit, compare the independently computed left and right results.

Fully worked example

Graphical or geometric meaning

A jump graph has two different approach heights at the same vertical line. Open or closed endpoint dots show actual inclusion, but the nearby branch, not the dot's fill, determines its one-sided limit.

Common mistakes and why they fail

Verification and reasonableness checks

• Use a tiny input on the correct side and see which branch applies.
• Compute the two sides independently before comparing.
• Confirm the approach remains in the function's domain.

The direction belongs to the input

In $\lim_{x\to a^-}f(x)$, the minus means inputs smaller than $a$, not negative outputs; $a^+$ similarly means inputs greater than $a$. For a piecewise function, use the formula active on that side. A two-sided limit exists only when both one-sided limits exist and agree as the same finite value, or the same infinite behavior under the course convention. The point value $f(a)$ does not affect this agreement. Tables must use inputs genuinely confined to one side, and graphs must distinguish open from closed points. At a domain endpoint, the available one-sided limit is the appropriate continuity test even though a two-sided limit is unavailable.

Practice

1. Find $\lim_{x\to0^+}\sqrt x$.
2. For $\operatorname{sgn}(x)$, find the left limit at zero.
3. When does a two-sided limit exist?

Connections and next steps

• Limits
• Continuity
• Infinite Limits