Question

${lim}_{x\to \infty }\frac{|x|}{x}$ is equal to

• A. 1
• B. -1
• C. 0
• D. Does not exist
  

Answer 1

The correct option is A

1

We know that,
$|x|=\{\begin{array}{cc}x,& Ifx\ge 0\\ -x,& Ifx<0\end{array}$
$\therefore \frac{|x|}{x}=\{\begin{array}{cc}\frac{x}{x},& ifx\ge 0\\ \frac{-x}{x},& ifx<0\end{array}=\{\begin{array}{cc}1,& ifx\ge 0\\ -1,.& ifx<0\end{array}$
Now, for all $x\ge 0$ (hiowever, x may large be),
$\frac{|x|}{x}=1$
$\therefore {lim}_{x\to \infty }\frac{|x|}{x}=1$