Question

How do I find the limit of an exponential function?

Answer 1

The limit of an exponential function is equal to the limit of the exponent with the same base. It is called the limit rule of an exponential function. We will use the property of limits such as: $\mathop {\lim }\limits_{x \to a} {b^{f(x)}}\, = \,{b^{\mathop {\lim }\limits_{x \to a} f(x)}}$

Complete step by step answer:
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 ${b^{f(x)}}$ in mathematics.
The limit of the exponential function ${b^{f(x)}}$ as $x$ approaches $a$ is written in the following mathematical form.
$\mathop {\lim }\limits_{x \to a} {b^{f(x)}}$
It is equal to the limit of the function $f(x)$as $x$ approaches $a$ with the base $b$
$\Rightarrow \,\mathop {\lim }\limits_{x \to a} {b^{f(x)}}\, = \,{b^{\mathop {\lim }\limits_{x \to a} f(x)}}$
It is called the limit rule of an exponential function in calculus.
Example: Evaluate $\mathop {\lim }\limits_{x \to 1} \,({3^{x + 2}})$
Firstly we find the value of the given function by directly putting the value of the limit.
$\Rightarrow \mathop {\lim }\limits_{x \to 1} \,({3^{x + 2}})\, = \,{3^{1 + 2}}$
$\Rightarrow {3^3}\, = 27$
Now we will evaluate ${3^{\mathop {\lim }\limits_{x \to 1} (x + 2)}}$by directly putting the value of the limit.
$\Rightarrow {3^{\mathop {\lim }\limits_{x \to 1} (x + 2)}}\, = {3^{1 + 2}}$
$\Rightarrow {3^3}\, = \,27$
Therefore, it is evaluated that $\mathop {\lim }\limits_{x \to 1} ({3^{x + 2}})\, = \,{3^{\mathop {\lim }\limits_{x \to 1} (x + 2)}}\, = \,27$

Note:
We can calculate the limit of an exponential function by direct substitution of limit in the function or by limit rule of an exponential function the results are the same for both the cases since the limit does not involve any indeterminate form.