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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2016 · Mathematics · Continuity and Differentiability

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 of the following statements is correct?

Options

  1. ABoth f(x)f(x) and g(x)g(x) are continuous at x=0x=0.
  2. Bf(x)f(x) is continuous at x=0x=0, but g(x)g(x) is not continuous at x=0x=0.
  3. Cg(x)g(x) is continuous at x=0x=0, but f(x)f(x) is not continuous at x=0x=0.
  4. DBoth f(x)\mathrm{f}(\mathrm{x}) and g(x)\mathrm{g}(\mathrm{x}) are discontinuous at x=0\mathrm{x}=0.

Correct answer

C. g(x)g(x) is continuous at x=0x=0, but f(x)f(x) is not continuous at x=0x=0.

Step-by-step solution

f(x)=[x] and  g(x)=sin⁡xlim⁡x→0+f(x)=[0+h]=0lim⁡x→0−f(x)=[0−h]=−1f(0)=0\begin{aligned} & f(x)=[x] \text{\text{ and } } g(x)=\sin x \\ & \lim _{x \rightarrow 0^{+}} f(x)=[0+h]=0 \\ & \lim _{x \rightarrow 0^{-}} f(x)=[0-h]=-1 \\ & f(0)=0 \end{aligned}\Rightarrow f(x)isnotcontinuousatis not continuous atx=0andalsoand alsog(x)iscontinuousatis continuous atx=0$. (every trignometric function is continuous).
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