The literals $a$ and $b$ are two constants, and $x$ is a variable. A function in terms of $x$ is denoted by $f(x)$ in mathematics. An exponential function in terms of constant $b$ and function $f(x)$ is written as $b^{\displaystyle f(x)}$ mathematically. In this case, $a$ is a value of the variable $x$.

Now, let’s start deriving the limit rule of an exponential function in calculus mathematically.

The limit of a function $f(x)$ as the input $x$ approaches $a$ is written as follows.

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

Take, the limit of the function as $x$ tends to $a$ is equal to $L$.

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

Now, find the limit of the function as $x$ closer to $a$ by direct substitution method.

$\implies$ $L \,=\, f(a)$

Now, write the limit of an exponential function as $x$ approaches $a$ in mathematical form.

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

Use the direct substitution method and evaluate the limit of the exponential function as $x$ approaches $a$.

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

According to the first step, $L = f(a)$. So, replace the value of $f(a)$ by $L$ in the above mathematical equation.

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

Actually, it is taken that $L \,=\, \displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$ in the first step.

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

Therefore, it is proved that the limit of an exponential function is equal to the limit of the exponent with same base. Thus, the limit rule of an exponential function is proved mathematically in calculus.

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