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NDA 2020
Exam
NDA
Year
2020
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2020 · Mathematics · Continuity and Differentiability

If f(x)=( x)/(x) , where x ∈ ℝ , is to be continuous at x=0 , then...

If f(x)=sin⁡xxf(x)=\frac{\sin x}{x}, where x∈Rx \in \mathbb{R}, is to be continuous at x=0x=0, then the value of the function at x=0x=0

Options

  1. Ashould be 0
  2. Bshould be 1
  3. Cshould be 2
  4. Dcannot be determined

Correct answer

B. should be 1

Step-by-step solution

As the given function is continuous at x=0x=0, then LHL=RHLlim⁡x→0−f(x)=lim⁡x→0−sin⁡xx=lim⁡x→0−[cos⁡x1]=1lim⁡x→0+f(x)=lim⁡x→0+sin⁡xx=lim⁡x→0+[cos⁡x1]=1∴sin⁡xx\mathrm{LHL}=\mathrm{RHL}\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}} \frac{\sin x}{x}=\lim _{x \rightarrow 0^{-}}\left[\frac{\cos x}{1}\right]=1\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} \frac{\sin x}{x}=\lim _{x \rightarrow 0^{+}}\left[\frac{\cos x}{1}\right]=1\therefore \frac{\sin x}{x} at x=0x=0, should be 1.
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