Composite Limit rule

Formula

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$ $\,=\,$ $f{\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}\Big)$

Introduction

Suppose $f(x)$ and $g(x)$ represent two functions in terms of $x$. The composition of them is expressed as $f{\Big(g{(x)}\Big)}$ in mathematics. In calculus, the limit of a composite function as $x$ approaches $a$ is written as follows.

$\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$

The limit of $f$ of $g$ of $x$ as $x$ closer to $a$ is equal to $f$ of the limit of $g$ of $x$ as $x$ tends to $a$.

$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{\Big(g{(x)}\Big)}}$ $\,=\,$ $f{\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}\Big)$

This mathematical equation is called the composite limit rule and it is used to find the limit of a function, which is formed by the composition of two or more functions.

Example

Let’s verify the composite limit rule from the following understandable example problem.

Evaluate $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \cos{\big(x^2-4\big)}}$

Now, find the limit of the cosine of angle $x$ square minus $4$ as $x$ approaches $2$ by the direct substation.

$=\,\,\,$ $\cos{\big((2)^2-4\big)}$

$=\,\,\,$ $\cos{(4-4)}$

$=\,\,\,$ $\cos{(0)}$

$=\,\,\,$ $1$

Similarly, find the cosine of the limit of the function $x$ squared minus $4$ as $x$ closer to $2$ by the direct substitution method.

$=\,\,\,$ $\cos{\Big(\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \big(x^2-4\big)\Big)}}$

$=\,\,\,$ $\cos{\Big(\big((2)^2-4\big)\Big)}$

$=\,\,\,$ $\cos{(4-4)}$

$=\,\,\,$ $\cos{(0)}$

$=\,\,\,$ $1$

$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \cos{\big(x^2-4\big)}}$ $\,=\,$ $\cos{\Big(\displaystyle \large \lim_{x \,\to\, 2}{\normalsize \big(x^2-4\big)\Big)}}$ $\,=\,$ $1$

Proof

Learn how to derive the composition limit rule to find the limit rule of a composite function.