Constant multiple rule of Limits

Formula

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

The limit of product of a constant and a function is equal to product of that constant and limit of the function. This limit property is called as constant multiple rule of limits.

Introduction

In calculus, the limit of product of a constant and a function has to evaluate as the input approaches a value. It is often appeared in limits. So, it is very important to know how to deal such functions in mathematics.

$x$ is a variable, and $k$ and $a$ are constants. The function in terms of $x$ is represented by $f{(x)}$. The limit of product of a constant ($k$) and the function $f{(x)}$ as the input $x$ approaches a value $a$ is written mathematically as follows.

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

It is equal to the product of the constant and the limit of the function.

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

It is called the constant multiple rule of limits in calculus.

Proof

Learn how to derive the constant multiple property of limits in calculus.