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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Limits

NDA 2016 · Mathematics · Limits

Paragraph: Let f(x)=[x] , where [.] is the greatest integer...

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

Options

  1. Alim⁡x→0(fog)(x)\lim _{x \rightarrow 0}(f o g)(x) exists
  2. Blim⁡x→0(\lim _{x \rightarrow 0}( gof )(x))(x) exists
  3. Clim⁡x→0+(f∘g)(x)=lim⁡x→0−(\lim _{x \rightarrow 0+}(f {}^{\circ} g)(x)=\lim _{x \rightarrow 0-}( gof )(x))(x)
  4. Dlim⁡x→0+(fog)(x)=lim⁡x→0+(gof)(x)\lim _{x \rightarrow 0+}(f o g)(x)=\lim _{x \rightarrow 0+}(g o f)(x)

Correct answer

D. lim⁡x→0+(fog)(x)=lim⁡x→0+(gof)(x)\lim _{x \rightarrow 0+}(f o g)(x)=\lim _{x \rightarrow 0+}(g o f)(x)

Step-by-step solution

(f∘g)(x)=[sin⁡x]lim⁡x→0+(fog)(x)=lim⁡x→0+[sin⁡x]=[h] where h>0⇒lim⁡x→0+(fog)(x)=0lim⁡x→0−(f∘g)(x)=lim⁡x→0−[sin⁡x]=[h] where h<0⇒lim⁡x→0−(fog)(x)=−1⇒lim⁡x→0(fog)(x) does not exist. \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 > 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 < 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, (gof)(x)=sin⁡[x](g o f)(x)=\sin [x]lim⁡x→0+(gof)(x)=lim⁡x→0+sin⁡[x]=sin⁡0=0lim⁡x→0−(gof)(x)=lim⁡x→0−sin⁡[x]=sin⁡(−1)=−0.01745⇒lim⁡x→0(gof)(x) does not exist. ⇒lim⁡x→0+(fog)(x)=lim⁡x→0+(gof)(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}$$
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