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

It is a limit rule for finding the limit of an exponential function whose base is a constant and exponent is a function.

$b$ is a constant. $x$ is a variable and $f{(x)}$ is a function in terms of variable $x$.

The constant $b$ and function $f{(x)}$ are formed an exponential function.

$b^{\displaystyle f{(x)}}$

The limit of this exponential function as x approaches a is written as follows.

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

The limit of exponential function can be evaluated by using power rule of limit.

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

The base $b$ is a constant and it does not have at least one variable in it but the exponent $f{(x)}$ is a function.

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

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

Therefore, it is proved in calculus that the limit of an exponential function whose base is a constant is equal to constant is raised to the power of limit of the exponent (function). The limit property is called as constant base power rule of limit.

List of most recently solved mathematics problems.

Jul 04, 2018

Limit (Calculus)

Evaluate $\displaystyle \large \lim_{x \,\to\, \tan^{-1}{3}} \normalsize {\dfrac{\tan^2{x}-2\tan{x}-3}{\tan^2{x}-4\tan{x}+3}}$

Jun 23, 2018

Limit (Calculus)

Evaluate $\displaystyle \large \lim_{x \to 0} \normalsize \dfrac{e^{x^2}-\cos{x}}{x^2}$

Jun 22, 2018

Integral Calculus

Evaluate $\displaystyle \int \dfrac{1+\cos{4x}}{\cot{x}-\tan{x}} dx$

Jun 21, 2018

Limit

Evaluate $\displaystyle \large \lim_{x \to \infty} \normalsize {\sqrt{x^2+x+1}-\sqrt{x^2+1}}$

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.