Q73 / 110 · VITEEE 2022 · Mathematics

If \(\lim _{x \rightarrow 0} \frac{((a-n) n x-\tan x) \sin n x}{x^2}=0\), where \(n\) is non-zero real number, then a is equal to

  • A
    0
  • B
    \(\frac{\mathrm{n}+1}{\mathrm{n}}\)
  • C
    \(\mathrm{n}\)
  • Correct answer
    D
    \(\mathrm{n}+\frac{1}{\mathrm{n}}\)

Solution

\(\begin{aligned} & \lim _{\mathrm{x} \rightarrow 0} \mathrm{n} \frac{\sin \mathrm{nx}}{\mathrm{nx}} \cdot \lim _{\mathrm{x} \rightarrow 0}\left((\mathrm{a}-\mathrm{n}) \mathrm{n}-\frac{\tan \mathrm{x}}{\mathrm{x}}\right)=0 \\ & \Rightarrow \mathrm{n}((\mathrm{a}-\mathrm{n}) \mathrm{n}-1)=0 \\ & \Rightarrow(\mathrm{a}-\mathrm{n}) \mathrm{n}=1 \Rightarrow \mathrm{a}=\mathrm{n}+\frac{1}{\mathrm{n}} . \end{aligned}\)

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

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