Properties of Limits

There are six fundamental properties in limits. They are used as formulas in some basic operations and also used in evaluating limits of the functions in calculus.

The limit of sum of two or more functions 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 difference of two functions is equal to difference 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 product of two or more functions 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 is equal to quotient of their limits.

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

Exponentiation

The limit of an exponential function is equal to exponentiation of their limits.

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

Composition

The limit of composition of two functions is equal to the value of the function for the limit of its internal function.

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

Formulas

List of standard results of limits with proofs to use them as formulas in calculus.

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