Let [t] denote the greatest integer
Question:

Let $[t]$ denote the greatest integer $\leq t$ and $\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=\mathrm{A}$. Then the function, $f(x)=\left[x^{2}\right] \sin (\pi x)$ is discontinuous, when $x$ is equal to :

1. (1) $\sqrt{\mathrm{A}+1}$

2. (2) $\sqrt{\mathrm{A}+5}$

3. (3) $\sqrt{\mathrm{A}+21}$

4. (4) $\sqrt{\mathrm{A}}$

Correct Option: 1

Solution:

$\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=A \Rightarrow \lim _{x \rightarrow 0} x\left[\frac{4}{x}-\left\{\frac{4}{x}\right\}\right]=A$

$\Rightarrow \quad \lim _{x \rightarrow 0} 4-x\left\{\frac{4}{x}\right\}=A \Rightarrow 4-0=\mathrm{A}$

As, $f(x)=\left[x^{2}\right] \sin (\pi x)$ will be discontinuous at nonintegers

And, when $x=\sqrt{A+1} \Rightarrow x=\sqrt{5}$, which is not an integer.

Hence, $f(x)$ is discontinuous when $x$ is equal to

$\sqrt{A+1}$