One-Sided Limit
A one-sided limit looks at what value a function approaches from only one direction, either from the left or from the right.
Formula
\lim_{x \to a^-} f(x) \text{ and } \lim_{x \to a^+} f(x)
Definition
A one-sided limit checks what a function is heading toward when you approach a point from just one side, either from the left (smaller numbers) or from the right (larger numbers). The left-hand limit $\lim_{x \to a^-} f(x) = L$ means $f(x)$ approaches $L$ as $x$ approaches $a$ through values less than $a$; the right-hand limit $\lim_{x \to a^+} f(x) = M$ means $x$ approaches from above. Formally, $\lim_{x \to a^-} f(x) = L$ if for every $\epsilon > 0$ there exists $\delta > 0$ such that $a - \delta < x < a$ implies $|f(x) - L| < \epsilon$, with the analogous statement on the right. The two-sided limit exists only if both one-sided limits are equal, and a function is continuous at $a$ if and only if both one-sided limits equal $f(a)$.
Example
Think of a cliff edge: walking toward it from the left, you approach a drop; walking from the right, you approach it from open air, so the view differs depending on which side you come from. For $f(x) = |x|/x$, the left-hand limit as $x \to 0$ is $-1$ and the right-hand limit is $+1$; since they differ, $\lim_{x \to 0} f(x)$ does not exist. The floor function $\lfloor x \rfloor$ shows the same idea: $\lim_{x \to 2^-} \lfloor x \rfloor = 1$ while $\lim_{x \to 2^+} \lfloor x \rfloor = 2$, a jump discontinuity precisely characterized by unequal one-sided limits.
Key Insight
Sometimes a function behaves differently on each side of a point, so checking both sides separately tells the full story, and comparing one-sided limits is the standard technique for determining whether a two-sided limit exists at a break or corner in a graph. One-sided limits are also essential in distributional analysis, Laplace transforms, and piecewise-defined differential equations where domain boundaries carry physical meaning.