Q227 / 270 · NDA 2023 (Phase 1) · Mathematics

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Question: What is $\lim _{x \rightarrow 0} \frac{f(x)}{x}$ equal to?

  • A
    $-1$
  • Correct answer
    B
    $0$
  • C
    $1$
  • D
    $2$

Solution

Solving $f(x)$ $\begin{aligned} f(x)=\left(x^2-2 x^2\right) \cos x-x( & 2 \sin x-2 x \tan x) \\ & +1\left(2 x \sin x-x^2 \tan x\right)\end{aligned}$ $=-x^2 \cos x-2 x \sin x+2 x^2 \tan x+2 x \sin x-x^2 \tan x$ $f(x)=x^2 \tan x-x^2 \cos x=x^2(\tan x-\cos x)$ $\therefore \quad \lim _{x \rightarrow 0} \frac{f(x)}{x}=\lim _{x \rightarrow 0} \frac{x^2(\tan x-\cos x)}{x}$ $=\lim _{x \rightarrow 0} x(\tan x-\cos x)=0$

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

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