Math Doubts

Constant multiple Law of Limit

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.

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



Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more