# 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.

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