Question

$\underset{x\to \infty }{\lim }\frac{\left|x\right|}{x}$ is equal to

(a) 1 (b) −1 (c) 0 (d) does not exist

Answer 1

We know that,

$$\begin{gathered} \left|x\right|=\left\{\begin{array}{ll}x,& \mathrm{if}x\ge 0\\ -x,& \mathrm{if}x<0\end{array}\right. \\ \therefore \frac{\left|x\right|}{x}=\left\{\begin{array}{ll}\frac{x}{x},& \mathrm{if}x\ge 0\\ \frac{-x}{x},& \mathrm{if}x<0\end{array}\right.=\left\{\begin{array}{ll}1,& \mathrm{if}x\ge 0\\ -1,& \mathrm{if}x<0\end{array}\right. \end{gathered}$$

Now, for all x ≥ 0 (however, x may large be), $\frac{\left|x\right|}{x}=1$.

$\therefore \underset{x\to \infty }{\lim }\frac{\left|x\right|}{x}=1$

Hence, the correct answer is option (a).