Limits of Exponential functions

To find limits of exponential functions, it is essential to study some properties and standards results in calculus and they are used as formulas in evaluating the limits of functions in which exponential functions are involved.

Properties

There are four basic properties in limits, which are used as formulas in evaluating the limits of exponential functions.

Power Rule

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

Constant Base Power Rule

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

Constant Exponent Power Rule

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

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

Standard Results

There are five standard results in limits and they are used as formulas while finding the limits of the functions in which exponential functions are involved.

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

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

$(3) \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{a^{\displaystyle \normalsize x}-1}{x}}$ $\,=\,$ $\log_{e}{a}$

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

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

Problems

List of solved limits problems for evaluating the limits of functions in which exponential functions are involved.

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