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.

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