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

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.

$x$ 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.

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

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

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

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.

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

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

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.

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

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