Q104 / 160 · AP EAMCET 2021 (25 Aug Shift 2) · Mathematics

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

  • A
    1
  • Correct answer
    B
    0
  • C
    -1
  • D
    does not exist

Solution

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

Hence, the correct answer is option: 'B'.

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