# Limit Rule of an Exponential function

## Formula

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

The limit of an exponential function is equal to the limit of the exponent with same base. It is called the limit rule of an exponential function.

### Introduction

Let $a$ and $b$ represent two constants, and $x$ represents a variable. A function in terms of $x$ is written as $f(x)$ mathematically. In this case, the constant $a$ is a value of $x$, and the exponential function in terms of $b$ and $f(x)$ is written as $b^{\displaystyle f(x)}$ in mathematics.

The limit of exponential function $b^{\displaystyle f(x)}$ as $x$ approaches $a$ is written in the following mathematical form in calculus.

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

It is equal to the limit of the function $f(x)$ as $x$ approaches $a$ with the base $b$.

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

It is called the limit rule of an exponential function in calculus.

#### Example

Evaluate $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \Big(3^{\displaystyle x+2}\Big)}$

Find the value of the given function by direct substitution.

$\implies$ $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \Big(3^{\displaystyle x+2}\Big)}$ $\,=\,$ $3^{\displaystyle 1+2}$

$\implies$ $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \Big(3^{\displaystyle x+2}\Big)}$ $\,=\,$ $3^{\displaystyle 3}$

$\implies$ $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \Big(3^{\displaystyle x+2}\Big)}$ $\,=\,$ $27$

Now, evaluate $3^{\, \displaystyle \large \lim_{x \,\to\, 1}{\normalsize (x+2)}}$ by direct substitution.

$\implies$ $3^{\, \displaystyle \large \lim_{x \,\to\, 1}{\normalsize (x+2)}}$ $\,=\,$ $3^{(1+2)}$

$\implies$ $3^{\, \displaystyle \large \lim_{x \,\to\, 1}{\normalsize (x+2)}}$ $\,=\,$ $3^{3}$

$\implies$ $3^{\, \displaystyle \large \lim_{x \,\to\, 1}{\normalsize (x+2)}}$ $\,=\,$ $27$

Therefore, it is evaluated that $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \Big(3^{\displaystyle x+2}\Big)}$ $\,=\,$ $3^{\, \displaystyle \large \lim_{x \,\to\, 1}{\normalsize (x+2)}}$ $\,=\,$ $27$

### Proof

Learn how to derive the limit rule of an exponential function in calculus mathematically.

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