Math Doubts

Limit Rules of Exponential functions

There are nine types of limit rules in which exponential functions are involved in various forms and they are called as limits rules for exponential functions. The properties of exponential functions limit laws are used as formulas in calculus.

Power Rule

It is a limit rule for an exponential function in which both base and exponent are functions.

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

Constant Base Power Rule

It is a limit rule for an exponential function whose base is a constant.

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

If base of exponential function is a Napier’s mathematical constant $e$, then it is called as natural base power rule.

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

Constant Exponent Power Rule

A limit rule for an exponential function in which the base is a function and exponent is a constant.

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

Root Power Rule

A limit rule for an exponential function in which the power is a root.

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

Ratio subtraction Natural Exponential Limit rule

A limit rule for a ratio of subtraction of one from natural exponential function to variable.

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

Ratio subtraction Exponential Limit rule

A limit rule for a ratio of subtraction of one from exponential function to variable.

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

Subtractions Ratio Exponential Limit Rule

It is a limit rule for a function formed by the quotient of subtraction of exponential functions by the subtraction of bases of them.

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

Exponential Binomial Zero Limit Rule

It is a limit rule and used for a binomial function in exponential form as the limit value tends to zero.

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

Exponential Binomial Infinity Limit Rule

It is another limit rule and used for a binomial function in exponential form as the limit value approaches infinity.

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



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