If [.] here denotes the greatest integer function, $\lim _{x \rightarrow 0} x^7\left[\frac{1}{x^3}\right]$ is equal to

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Aicharya · Accepted Answer

$\lim _{x ightarrow 0} x^7\left[\frac{1}{x^3}ight]$ RHL Let $x=0+h$, where $h ightarrow 0$ $ \lim _{h ightarrow 0} h^7\left[\frac{1}{(0+h)^3}ight]=0 $ $\begin{aligned} & 	ext { LHL Let } x=0-h 	ext { where } h ightarrow 0 \ & \qquad \lim _{h ightarrow 0}(-h)^7\left[\frac{1}{(0-h)^3}ight] \ & \lim _{h ightarrow 0}-h^7 \cdot(-1)=0\end{aligned}$ $\begin{aligned} & \because \mathrm{LHL}=\mathrm{RHL}=0 \ & 	herefore \lim _{x ightarrow 0} x^7\left[\frac{1}{x^3}ight]=0\end{aligned}$

If [.] here denotes the greatest integer function, $\lim _{x \rightarrow 0} x^7\left[\frac{1}{x^3}\right]$…

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