1. Tardigrade
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  4. Let [t] denote the greatest integer le t displaystyle limx → 0 x[(4/x)]=A. Then the function. and f (x)=[x2]sin(π x) is discontinuous, when x is equal to :

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

JEE MainJEE Main 2020Continuity and Differentiability

Solution:

$A =\displaystyle \lim_{x\to0} x \left[\frac{4}{x}\right] = \displaystyle \lim _{x\to 0} x \left[\frac{4}{x}\right]-x\left\{\frac{4}{x}\right\} = 4$
$ƒ\left(x\right) = \left[x^{2}\right]sin\left(\pi x\right)$ will be discontinuous at nonintegers
$\therefore x = \sqrt{A+1} \,i.e. \sqrt{5}$