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

Common mistakes and why they fail