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.

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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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