What is the limit of \[{e^{ - x}}\] as \[x \to \infty \]?

Hint: In this problem, we need to find out the limit of \[{e^{ - x}}\] as \[x\] approaches to\[\infty \]. Here, A limit tells us the value that a function approaches as that function's inputs get closer and closer to some number. The idea of a limit is the basis of all calculus. The expression for the derivative is the same as the expression that we started with, that is, \[{e^{ - x}}\]. It means the slope is the same as the function value for all points on the graph. We plotted a graph with respect to the given question.

Complete step by step solution:
In the given problem,
The function is \[f(x) = {e^{ - x}}\] and limit as \[x\] approaches infinity, \[x \to \infty \]
\[\mathop {\lim }\limits_{x \to \infty } f(x) = \mathop {\lim }\limits_{x \to \infty } {e^{ - x}}\]
By applying the value of \[x\] as \[\infty \], we can get
\[\mathop {\lim }\limits_{x \to \infty } {e^{ - x}} = {e^{ - \infty }}\]
Since,\[{e^{ - \infty }} = 0\], we get
\[\mathop {\lim }\limits_{x \to \infty } {e^{ - x}} = 0\]
Therefore, the limit of \[{e^{ - x}}\] as\[x \to \infty \] is \[0\].

Additional information: We note that the phrase\[f(x)\] is a limit as \[x\] approaches infinity and the limit is the value that a sequence approaches as the index approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. Here, The expression for the derivative is the same as the expression that we started with; that is, \[{e^{ - x}}\] and the range of \[x\] is \[(0,\infty )\].

Note:
We can also estimate the value of the given exponential function by remembering the graph. The exponential function of \[f\] with base\[e\] is defined by \[f(x) = {e^{ - x}}\] is an equation in which the variable is an exponent,and the graph is exponentially decreasing w.r.t. $x$. Where, \[x\] is a real number and \[e\] is a positive constant. Here we have to plot a graph as follows with respect to the given equation.