Math Doubts

Subtraction Law of Limits

Formula

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

The limit of subtraction of two functions as the input approaches some value is equal to difference of their limits. It is called as subtraction rule of limits and also called as difference property of limits.

Proof

$x$ is a variable and two functions $f{(x)}$ and $g{(x)}$ are developed in terms of $x$. The limits of $f{(x)}$ and $g{(x)}$ as $x$ approaches $a$ are written mathematically as follows.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}$ $\,=\,$ $f{(a)}$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}$ $\,=\,$ $g{(a)}$

Limit of Subtraction of functions

Find the limit of subtraction of the functions $f{(x)}$ and $g{(x)}$ as $x$ tends to $a$. In this case, it is taken that the function $g{(x)}$ is subtracted from the function $f{(x)}$.

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

Evaluate Limit of Subtraction of functions

Substitute $x$ is equal to $a$ to find the limit of subtraction of the function as $x$ approaches $a$.

$\implies \displaystyle \large \lim_{x \,\to\, a}{\normalsize \Big[f{(x)}-g{(x)}\Big]}$ $\,=\,$ $f{(a)}-g{(a)}$

Replace the Limits of functions

Now, replace the values of $f{(a)}$ and $g{(a)}$ in limit form.

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

Therefore, it is proved that the limit of subtraction of two functions as input approaches some value is equal to difference of their limits. It is used as a formula in calculus to find limit of two functions which are involved in subtraction.



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