Q. 91 - Calculus

Question

Prove the second part of Theorem$1.31$(a): If $k>0$, then $\underset{x\to \infty }{\lim }{x}^{-k}=0$.

Step-by-Step Solution

Answer

It is proved that if $k>0$, then $\underset{x\to \infty }{\lim }{x}^{-k}=0$.

Step 1. Given Information

We are given that if $k>0$, then $\underset{x\to \infty }{\lim }{x}^{-k}=0$.

Step 2. Proving the statement

Consider the limit expression,

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

Hence Proved.