Math Doubts

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.

Math Doubts

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the maths problems in different methods with understandable steps.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved