Q. 92 - Calculus

Question

Prove the $k=1$ case of the first part of Theorem 1.31(b): that $\underset{x\to \infty }{\lim }{e}^x=\infty$. (Hint: Given $M>0$, choose $N=\ln M$. Then if $x>N=\ln M$, we must have $x=\ln M+c$ for some positive number c. Use this to show that $e^x>M$.)

Step-by-Step Solution

Answer

The first part of Theorem 1.31(b) is proved.

$\underset{x\to \infty }{\lim }{e}^x=\infty$

Step 1. Given Information

We are given a theorem 1.31(b),

$\underset{x\to \infty }{\lim }{e}^x=\infty$

Step 2. Proving the statement.

Proving the statement,

For $x>N$,

$$\begin{gathered} \mathrm{x}>\ln M \\ x=\ln M+c \\ {e}^x=e^{\ln M+c} \\ {e}^x=M\left(e^c\right) \\ {x}^k>M \end{gathered}$$

Since 'M' is arbitrary. This implies that $\underset{x\to \infty }{\lim }{e}^x=\infty$.

Hence Proved.