Q100 / 160 · AP EAMCET 2022 (05 Jul Shift 2) · Mathematics

If $a>0 .[\cdot]$ denotes greatest integer function and $\lim _{x \rightarrow a^{-}}\left(\frac{|x|^3}{a}-\left[\frac{x}{a}\right]^3\right)=k, \lim _{x \rightarrow a^{+}}\left(\frac{|x|^3}{a}-\left[\frac{x}{a}\right]^3\right)=l$, then

  • A
    $k=l$
  • Correct answer
    B
    $k-l=1$
  • C
    $l-k=1$
  • D
    $l=a^2, k$ does not exist

Solution

For $x \rightarrow a^{-}$ $\frac{x}{a} < 1 \Rightarrow\left[\frac{x}{a}\right]=0$ $\lim _{x \rightarrow a^{-}} \frac{|x|^3}{a}=\frac{a^3}{a}=a^2$ for $x \rightarrow a^{+}$ $\frac{x}{a}>1 \Rightarrow\left[\frac{x}{a}\right]=1$ $\lim _{x \rightarrow a^{+}} \frac{|x|^3}{a}=\frac{a^3}{a}=a^2$ $\begin{aligned} & k=a^2-0=a^2 \\ & l=\left(a^2-1\right) \\ & k-l=a^2-\left(a^2-1\right) \\ & =1\end{aligned}$ Note: In question, for $l$ also limit is $x \rightarrow a^{-}$which should $x \rightarrow a^{+}$

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

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