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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2019 · Mathematics · Continuity and Differentiability

Consider the following statements in respect of the function f(x)=...

Consider the following statements in respect of the function f(x)=sin⁡f(x)=\sin\left(\frac{1}{x}\right)forx10andf(0)=0for x^1 0 and f(0)=0; 1. \lim _{x \rightarrow 0}f(x)f(x) exists 2. f(x)f(x) is continuous at x=0x=0 Which of the above statements is/are correct?

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

D. Neither 1 nor 2

Step-by-step solution

f(0)=0f(0)=0 and, f(x)=sin⁡f(x)=\sin\left(\frac{1}{x}\right) (i) \lim _{x \rightarrow 0}f(x)=\lim _{x \rightarrow 0} \sin(1x)\left(\frac{1}{x}\right) if x→0+;lim⁡x→0+sin⁡(1x)=sin⁡∞ifx → 0^{+} ; \lim _{x \rightarrow 0^{+}} \sin \left(\frac{1}{x}\right)=\sin \infty ifx → 0^{-} ; lim⁡x→0−sin⁡(1x)=−sin⁡∞∵sin⁡∞≠−sin⁡∞∴LHL≠RHL(ii)at\lim _{x \rightarrow 0^{-}} \sin \left(\frac{1}{x}\right)=-\sin \infty\because \sin \infty \neq-\sin \infty\therefore \mathrm{LHL} \neq \mathrm{RHL} (ii) atx=0, LHL≠RHL∴\mathrm{LHL} \neq \mathrm{RHL}\therefore f(x)isnotcontinous∴is not continous \therefore both statements are wrong.
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