\( x 0 tan x - x x 2 tan x \) is equal to:" - Step-by-step solution, answer key, and detailed explanation on MCA Prep."> \... - Solved | MCA Prep" /> \( x 0 tan x - x x 2 tan x \) is equal to:" - Step-by-step solution, answer key, and detailed explanation on MCA Prep." /> \... - Solved | MCA Prep" /> \( x 0 tan x - x x 2 tan x \) is equal to:" - Step-by-step solution, answer key, and detailed explanation on MCA Prep." />

Question 42

Mathematics Limit of Functions Hard

The value of&nbsp;<span class="math-tex">\(\rm \displaystyle\lim_{x\to 0}\dfrac{\tan x - x}{x^2 \tan x}\)</span>&nbsp;is equal to:

(A) 0
(B) 1
(C) <span class="math-tex">\(\dfrac{1}{2}\)</span>
(D) <span class="math-tex">\(\dfrac{1}{3}\)</span>
✓ Correct Answer: Option D Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

For small \(x\): \(\tan x = x + \frac{x^3}{3} + \dots\), so \(\tan x - x \approx \frac{x^3}{3}\), and \(x^2\tan x \approx x^3\).

\(\lim_{x \to 0} \frac{\tan x - x}{x^2\tan x} = \lim_{x \to 0} \frac{x^3/3}{x^3} = \frac{1}{3}\)

Answer: (D) \(\frac{1}{3}\)

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