Constant multiple Law 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 as the input approaches some value, is equal to product of constant and the limit of the function. The limit property is called as constant multiple rule of limit.

Proof

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

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

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 product of constant and limit of the function.

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