Q. 93 - Calculus

Question

Prove the $k=1$ case of the second part of Theorem $1.31$(b): that $\underset{x\to \infty }{\lim }{e}^{-x}=0$.

Step-by-Step Solution

Answer

It is proved for theorem $1.31\left(b\right)$ that$\underset{x\to \infty }{\lim }{e}^{-x}=0$.

Step 1. Given Information

We are given,

$\underset{x\to \infty }{\lim }{e}^{-x}=0$

Step 2. Proving the statement

The limit is given by,

$$\begin{gathered} \underset{x\to \infty }{\lim }{e}^{-x}=\underset{x\to \infty }{\lim }\frac{1}{e^x} \\ =\frac{1}{e^{\infty }} \\ =\frac{1}{\infty } \\ =0 \end{gathered}$$

Hence Proved.