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

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.

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