Question

Use Definition 9 to prove that \(\mathop {{\rm{lim}}}\limits_{x \to \infty } {e^x} = \infty \).

Short answer

It is proved that \(\mathop {{\rm{lim}}}\limits_{x \to \infty } {e^x} = \infty \).

Step-by-step solution

Precise Definition of an infinite limit at infinity  

Assume \(f\) be any function defined on the interval \(\left( {a,\infty } \right)\). Then \(\mathop {{\rm{lim}}}\limits_{x \to - \infty } f\left( x \right) = \infty \),

which implies that for every \(M > 0\) there exist a positive number \(N\) such that if \(x > N\) then \(f\left( x \right) > M\).

Given \(M > 0\), we have to find \(N\) such that if \(x > N\) then \({e^x} > M\).

Simplify the inequality

Simplify the inequality \({e^x} > M\) for \(x\).

\(\begin{array}{c}{e^x} > M\\{\rm{ln}}{e^x} > {\rm{ln}}M\\x > {\rm{ln}}M\end{array}\)

Choose \(N = {\rm{max}}\left( {{\rm{1,ln}}M} \right)\) which satisfy if \(x > {\rm{max}}\left( {{\rm{1,ln}}M} \right)\) then \({e^x} > M\).

By using Precise Definition of an infinite limit at infinity, \(\mathop {{\rm{lim}}}\limits_{x \to \infty } {e^x} = \infty \).

Hence, it is proved that \(\mathop {{\rm{lim}}}\limits_{x \to \infty } {e^x} = \infty \).