Math Doubts

Proof of Constant multiple rule of Limits

$a$ and $k$ are two constants, and $x$ is a variable. $f(x)$ is a function in $x$. The limit of product of a constant $k$ and the function $f(x)$ as the input $x$ approaches a value $a$ is written as the following mathematical form.

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

It is called the constant multiple rule of limits and its equivalent value can be derived mathematically in three simple steps.

Use Product Rule of Limit

Consider the constant ($k$) as a function, then apply the product rule of limits for both functions.

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

Property of Constant function

The function $f{(x)}$ is defined in terms of $x$ but the constant function ($k$) does not contain at least one variable $x$. Therefore, the limit of constant function remains same mathematically.

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

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

Product Rule of Limit

Therefore, it has proved that the limit of product of constant and a function as the input tends to a value, is equal to the product of constant and the limit of the function.

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

Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more