Operations of Limits

The limits are involved in four mathematical operations. So, it is essential for everyone to know how to perform operations with limits.

Addition

The limit of sum of two or more functions as input approaches some value, is equal to sum of their limits.

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

Subtraction

The limit of subtraction of two functions as input approaches some value, is equal to subtraction of their limits.

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

Multiplication

The limit of multiplication of two or more functions as input tends to some value, is equal to product of their limits.

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

Division

The limit of quotient of two functions as input tends to some value, is equal to quotient of their limits.

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

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