Limit Laws for Functions¶

Theorem¶

Let $f\colon X \to\mathbb{R}$ and $g\colon X \to\mathbb{R}$ be functions that are defined on some punctured neighbourhood of $x_0\text{.}$ Suppose that $\lim\limits_{x\to x_0}f(x)=L\text{,}$ and $\lim\limits_{x\to x_0}g(x)=M$ where $L$ and $M$ are real numbers. Then, we have:

\begin{aligned} \lim_{x\to x_0}(f\pm g)(x)&=L\pm M,\\ \lim_{x\to x_0}\max\{f,g\}(x)&=\max\{L, M\},\\ \lim_{x\to x_0}\min\{f,g\}(x)&=\min\{L, M\},\\ \lim_{x\to x_0}(fg)(x)&=LM,\\ \lim_{x\to x_0}(f/g)(x)&=\frac{L}{M}\quad\text{if }M\neq0. \end{aligned}

Interpretation¶

Limit laws describe how limits at a point interact with algebraic operations on functions: the limit of a sum, difference, product, maximum, minimum or quotient (with a nonzero limit in the denominator) is obtained by applying the same operation to the separate limits. They allow limits of complicated expressions to be evaluated from the limits of simpler components.

In practice the laws are applied step by step. A polynomial is built from constants and $x$ by sums and products, so $\lim_{x\to x_0}p(x)=p(x_0)$ for every polynomial $p$ ; by the quotient law the same holds for a rational function at every point where its denominator is nonzero. In the language of continuity: sums, products and quotients of continuous functions are continuous wherever they are defined.

The laws work in one direction only. If both limits exist, so does the limit of the combination, but the combination may have a limit when the parts do not: the signum function $\operatorname{sgn}$ and its negative $-\operatorname{sgn}$ have no limit at $0$ , yet their sum is the zero function. The laws also do not cover infinite limits; a form such as $\infty-\infty$ has to be rewritten before any law applies.

A common slip is to apply the quotient law when the denominator tends to $0$ . For $\lim_{x\to1}\frac{x^2-1}{x-1}$ both parts tend to $0$ , the law does not apply, and $0/0$ is not a value. Cancelling the factor $x-1$ first gives $\lim_{x\to1}(x+1)=2$ $2$ . Such $0/0$ cases return later as indeterminate forms.

The same laws hold for one-sided limits and limits at infinity, and they are the function version of the limit laws for sequences.

Key Results¶

Limit laws apply only when both limits exist and are real numbers; they do not cover infinite limits or limits that fail to exist.
Algebraic combinations preserve convergence.
Division requires nonzero limiting denominators.

Examples¶

$\displaystyle \lim_{x\to 0}(x^2+\sin x)=0\text{.}$
$\displaystyle\lim_{x\to 1}\frac{x^2+1}{x+1}=1\text{.}$

Prerequisites¶

Limit at a Point

Sources¶

Math I: Calculus · Lecture 5

Symbols: infinity, real numbers

Last updated: 20 October 2026 03:02