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 :
Correct Option: 1
$\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=A \Rightarrow \lim _{x \rightarrow 0} x\left[\frac{4}{x}-\left\{\frac{4}{x}\right\}\right]=A$
$\Rightarrow \quad \lim _{x \rightarrow 0} 4-x\left\{\frac{4}{x}\right\}=A \Rightarrow 4-0=\mathrm{A}$
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
$\sqrt{A+1}$
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