Question

Find the limit \[\mathop {\lim }\limits_{x \to \infty } {e^x}\]

Answer 1

Hint: Function ${e^x}$ is the natural exponential function. The constant $e = 2.71828....$ is the unique base for which the constant of proportionality is 1.
Meaning of $x \to a$: Let x be a variable and be a constant. If x takes values closer and closer to a, then we say that x approaches to ‘a’ and we write it as $x \to a$.

Complete step-by-step solution:
We have given a function $f\left( x \right) = y = {e^x}$. The range of ${e^x}$ is$\left( {0,\infty } \right)$.
To find the limit x tends to infinity of the given function. We have to substitute numbers at the place of x. Then will find the value of a function ${e^x}$. We will use the value$e = 2.71828....$.
First, we will substitute 1.
${e^1} = e$
${e^2} = {\left( {2.7182} \right)^2} = 7.3890$
${e^3} = {\left( {2.71828} \right)^3} = 20.0855$
Similarly, we can substitute other values, and so on. From this, we can understand that when x goes infinitely large, the function grows infinitely large. Means as $x \to \infty $ then functions ${e^x} \to \infty $.
Hence, the limit does not exist because as x increases without bound, ${e^x}$ also increases without bound.
$\therefore \mathop {\lim }\limits_{x \to \infty } {e^x} = \infty $

Note: The number $e$ sometimes called the Euler’s number. Like the constant $\pi ,e$ is irrational and it is transcendental.
The function $f\left( x \right) = {e^x}$ is a function that is very important in calculus. It appears in many applications.
The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.
Exponential function: Any function in which an independent variable is in the form of an exponent; they are the inverse functions of logarithms.
Exponential functions are used to model populations, carbon date artifacts, help coroners determine the time of death, compute investments, as well as many other applications.