A limit describes the value a function approaches as the input approaches a point (or infinity), independent of whether the function equals that value at the point.
Notation and One-Sided Limits
\lim_{x \to a} f(x) = L means $f(x)$ approaches $L$ as $x$ approaches $a$ from both sides.
- \lim_{x \to a^-} f(x) — left-hand limit
- \lim_{x \to a^+} f(x) — right-hand limit
The two-sided limit exists only if both one-sided limits exist and are equal.
Continuity at a Point
$f$ is continuous at $x = a$ if:
- $f(a)$ is defined
- \lim_{x \to a} f(x) exists
- \lim_{x \to a} f(x) = f(a)
Polynomials and rational functions (away from zeros of the denominator) are continuous on their domains.
Useful Standard Limits
\lim_{x \to 0} \frac{\sin x}{x} = 1 \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \lim_{x \to 0} \frac{e^x - 1}{x} = 1Indeterminate Forms and L'Hôpital
Forms such as $0/0$ and $\infty/\infty$ are indeterminate: the limit may exist, be infinite, or fail to exist. L'Hôpital's rule applies when $f$ and $g$ are differentiable near $a$ (except possibly at $a$) and the quotient of derivatives has a limit:
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}Convert other indeterminate forms ($0·∞$, $∞−∞$, $1^∞$, …) into a quotient before applying the rule.
Worked Example
\lim_{x \to 0} \dfrac{1 - \cos x}{x^2} is $0/0$. Differentiate numerator and denominator:
\lim_{x \to 0} \dfrac{\sin x}{2x} = \dfrac{1}{2} (or apply the standard limit $(1-\cos x)/x^2 = 1/2$ directly).
Tools
Limits Calculator · Limits & Continuity Cheatsheet · Derivative Rules