Limits rules

The list of fundamental properties and standard results of limits.

Operations

There are four fundamental limit operations in calculus. Let’s learn each limit operation with proof and examples to understand how to use them in mathematics.

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

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

$(3).\,\,$ $\displaystyle \large \lim_{x\,\to\,a}{\normalsize \Big(f(x) \times g(x)\Big)}$ $\,=\,$ $\displaystyle \large \lim_{x\,\to\,a}{\normalsize f(x)}$ $\times$ $\displaystyle \large \lim_{x\,\to\,a}{\normalsize g(x)}$

$(4).\,\,$ $\displaystyle \large \lim_{x\,\to\,a}{\normalsize \dfrac{f(x)}{g(x)}}$ $\,=\,$ $\dfrac{\displaystyle \large \lim_{x\,\to\,a}{\normalsize f(x)}}{\displaystyle \large \lim_{x\,\to\,a}{\normalsize g(x)}}$

Likewise, there are two other limit operations in mathematics. So, let us learn them with proofs and example to learn how to use them mathematically in calculus.

$(5).\,\,$ $\displaystyle \large \lim_{x\,\to\,a}{\normalsize \dfrac{1}{f(x)}}$ $\,=\,$ $\dfrac{1}{\displaystyle \large \lim_{x\,\to\,a}{\normalsize f(x)}}$

$(6).\,\,$ $\displaystyle \large \lim_{x\,\to\,a}{\normalsize {f(x)}^{\displaystyle g(x)}}$ $\,=\,$ $\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize {f{(x)}\Big)}^{\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}}}$

Properties

List of fundamental properties of limits in mathematical form.

$(7) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize e^{f{(x)}}}$ $\,=\,$ $e^{\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}}$

$(8) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize {[f{(x)}]}^n}$ $\,=\,$ ${\Big[\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)}}\Big]}^n$

$(9) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \sqrt[\displaystyle n]{f{(x)}} }$ $\,=\,$ $\sqrt[\displaystyle n]{ \displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(x)} }}$

$(10) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f{(g{(x)})}}$ $\,=\,$ $f{\Big(\displaystyle \large \lim_{x \,\to\, a}{\normalsize g{(x)}}\Big)}$

Formulas

List of standard results in all branches of mathematics.

Exponential Limit Rules

Learn list of limit rules for exponential functions.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{e^{\displaystyle \normalsize x}-1}{x} \,=\, 1$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0} \normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x} \,=\, \log_{e}{a}$

$(3) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a} \dfrac{x^n-a^n}{x-a}$ $\,=\,$ $n.a^{n-1}$

$(4) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0} {(1+x)}^\frac{1}{x}$ $\,=\,$ $e$

$(5) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, \infty} {\Big(1+\dfrac{1}{x}\Big)}^{\displaystyle x}$ $\,=\,$ $e$

Trigonometric Limit Rules

Learn list of limits of trigonometric functions in calculus.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin{x}}{x}} \,=\, 1$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\tan{x}}{x}} \,=\, 1$

Inverse Trigonometric Limit Rules

Learn list of inverse trigonometric limit rules, used in calculus.

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sin^{-1}{x}}{x}} \,=\, 1$

$(2) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\tan^{-1}{x}}{x}} \,=\, 1$

Logarithmic Limit Rule

$(1) \,\,\,\,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\log_{e}{(1+x)}}{x}} \,=\, 1$