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

The limit of quotient of two functions as the input approaches some value is equal to quotient of their limits. It is called as quotient rule of limits and also called as division property of limits.

$x$ is a variable and two functions $f{(x)}$ and $g{(x)}$ are derived in terms of $x$. The limits of $f{(x)}$ and $g{(x)}$ as $x$ approaches to $a$ can be written mathematically as follows.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $\,=\,$ $f{(a)}$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$ $\,=\,$ $g{(a)}$

Now, write limit of quotient of the functions $f{(x)}$ and $g{(x)}$ as $x$ tends to $a$ in mathematical form.

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

Evaluate the limit of division of the functions as $x$ tends to $a$ by replacing $x$ by $a$.

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

Lastly, substitute the limits $f{(a)}$ and $g{(a)}$ in limit form.

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

Therefore, it has proved that the limit of quotient of two functions as input approaches some value is equal to quotient of their limits. So, it is called as quotient rule of limits and also called as division property of limits.

Latest Math Topics

Jan 06, 2023

Jan 03, 2023

Jan 01, 2023

Dec 26, 2022

Dec 08, 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