Let $f(x)$ and $g(x)$ be two functions in terms of $x$. The composition of them forms a composite function $f{\Big(g{(x)}\Big)}$. The limit of the composite function as $x$ approaches $a$ is written as follows in mathematics.

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

Let us learn how to derive the limit rule to find the limit of the composite function.

The limit of the function $g(x)$ as the value of $x$ closer to $a$ is written in mathematical form as follows.

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

Letâ€™s use the direct substation method to find the limit of the function $g(x)$ as $x$ tends to $a$.

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

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

The limit of the composite function $f{\Big(g{(x)}\Big)}$ as $x$ approaches $a$ is expressed in the following form in calculus.

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

Use the direct substation method and find evaluate the limit of the composite function as $x$ closer to $a$.

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

In the previous step, we have evaluated the following mathematical equation.

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

In the first step, the limit of the function $g(x)$ is evaluated when the value of $x$ approaches $a$.

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

Now, substitute the value of g(a) in the above mathematical equation to prove a limit rule for finding the limit of a composite function.

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

Latest Math Topics

Nov 11, 2022

Nov 03, 2022

Jul 24, 2022

Jul 15, 2022

Latest Math Problems

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

Oct 24, 2022

Sep 30, 2022

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

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

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

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved