Question

If $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}1,\mathrm{when}\mathrm{x}\mathrm{is}\mathrm{rational}\\ 0,\mathrm{when}\mathrm{x}\mathrm{is}\mathrm{irrational}\end{array}\right.$, then $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)=?$

• A. $0$
• B. $1$
• C. $\frac{1}{2}$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option

Step 1: Solving $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)$when $x$ belongs to a rational number.

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}1,\mathrm{x}\in \mathrm{q}\\ 0,\mathrm{x}\notin \mathrm{q}\end{array}\right.$, where $\mathrm{q}$ represents a rational number.

Left-hand limit (LHL):$\underset{\mathrm{h}\to 0}{\lim }\mathrm{f}(0-\mathrm{h})=\mathrm{f}(-\mathrm{h})=1$

Right-hand limit (RHL): $\underset{\mathrm{h}\to 0}{\lim }\mathrm{f}(0+\mathrm{h})=\mathrm{f}\left(\mathrm{h}\right)=1$

$\therefore \underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)=1$

Step 2: Solving $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ when x belongs to an irrational number.

Left-Hand Limit(LHL) : $\underset{\mathrm{h}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)=0$

Right-Hand Limit(RHL):$\underset{\mathrm{h}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)=0$

$\therefore \underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)=0$

As $\underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ can either be $0$ or$1$

Therefore, the limit doesn't exist.

Hence, the correct option is an option(D).