Q232 / 270 · NDA 2016 (Phase 2) · Mathematics

Paragraph: Let $f(x)=[x]$, where $[$.$] is the greatest integer function and $g(x)=\sin x$ be two real valued functions over $R$.
Question: Which one of the following statements is correct?

  • A
    $\lim _{x \rightarrow 0}(f o g)(x)$ exists
  • B
    $\lim _{x \rightarrow 0}($ gof $)(x)$ exists
  • C
    $\lim _{x \rightarrow 0+}(f \circ g)(x)=\lim _{x \rightarrow 0-}($ gof $)(x)$
  • Correct answer
    D
    $\lim _{x \rightarrow 0+}(f o g)(x)=\lim _{x \rightarrow 0+}(g o f)(x)$

Solution

$\begin{aligned} & (f \circ g)(x)=[\sin x] \\ & \lim _{x \rightarrow 0^{+}}(f o g)(x)=\lim _{x \rightarrow 0^{+}}[\sin x]=[h] \text { where } h \gt 0 \\ & \Rightarrow \lim _{x \rightarrow 0^{+}}(f o g)(x)=0 \\ & \lim _{x \rightarrow 0^{-}}(f \circ g)(x)=\lim _{x \rightarrow 0^{-}}[\sin x]=[h] \text { where } h \lt 0 \\ & \Rightarrow \lim _{x \rightarrow 0^{-}}(f o g)(x)=-1 \\ & \Rightarrow \lim _{x \rightarrow 0}(f o g)(x) \text { does not exist. } \end{aligned}$ Now, $(g o f)(x)=\sin [x]$ $\begin{aligned} & \lim _{x \rightarrow 0^{+}}(g o f)(x)=\lim _{x \rightarrow 0^{+}} \sin [x]=\sin 0=0 \\ & \lim _{x \rightarrow 0^{-}}(g o f)(x)=\lim _{x \rightarrow 0^{-}} \sin [x]=\sin (-1)=-0.01745 \\ & \Rightarrow \lim _{x \rightarrow 0}(g o f)(x) \text { does not exist. } \\ & \Rightarrow \lim _{x \rightarrow 0^{+}}(f o g)(x)=\lim _{x \rightarrow 0^{+}}(g o f)(x) \end{aligned}$

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

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