Subtraction of Limits Rule

Formula

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

Introduction

An expression can be formed in mathematics by the subtraction of functions. The difference between functions may interrupt the process of evaluating the limit. So, a limit formula is required to find the limit of difference of functions and it is called subtraction of limits rule. It is also called difference of limits rule.

Let’s denote two functions as $f(x)$ and $g(x)$, whereas let’s assume that the function $g(x)$ is subtracted from another function $f(x)$ and their difference is written as $f(x)-g(x)$ in mathematics. Its limit as the value of variable $x$ tends to a value $a$ is written as follows.

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

According to difference law of limits in calculus, the limit of a difference is equal to difference between their limits. So, the limit of difference between functions $f(x)$ and $g(x)$ is equal to difference between the limits of $f(x)$ and $g(x)$ as $x$ approaches to $a$.

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

The above equation mathematically expresses the difference of limits theorem and it can be used as a formula in calculus.

Examples

Let’s understand the limit difference rule from an easy limit problem.

Now, let’s learn more about the limit of a difference law in calculus.

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