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

The limit of product of two or more functions as the input approaches some value is equal to product of their limits. It is called as product rule of limits and also called as multiplication property of limits.

$x$ is a variable and two functions $f{(x)}$ and $g{(x)}$ are defined in terms of $x$. The limits of $f{(x)}$ and $g{(x)}$ as $x$ tends to $a$ are expressed 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, express limit of product of the function $f{(x)}$ and $g{(x)}$ as $x$ approaches $a$ in mathematical form.

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

Find the limit of product of functions as $x$ approaches $a$ by substituting $x$ is equal to $a$.

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

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

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

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

Therefore, it is proved that the limit of product of two functions as input approaches some value is equal to product of their limits. It is called as product rule of limits and can also be called as multiplication property of limits.

Remember, the law of product rule of limits is not limited to two functions and it can used more than two functions.

$\,\,\, \therefore \,\,\,\,\,\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big[f{(x)}.g{(x)}.h{(x)}\ldots\Big]}$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $\times$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$ $\times$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize h{(x)}} \ldots$

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