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NDA 2018 Maths · Mathematics · Derivatives · medium

Lim x→∞(1+2/x)^x=e^2 2. lim x→0(tan x)/(x)=0 Which of the statements given above is/are correct?

Single correct option

Consider the following statements:
1. lim⁡x→∞(1+2x)x=e2\lim_{x\to\infty}\left(1+\frac2x\right)^x=e^2
2. lim⁡x→0tan⁡xx=0\lim_{x\to0}\frac{\tan x}{x}=0
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (A) 1 only

Statement 1: (1+2x)x=[(1+2x)x/2]2→e2\left(1+\frac2x\right)^x=\left[\left(1+\frac2x\right)^{x/2}\right]^2\to e^2, so it is correct. Statement 2: tan⁡xx→1\frac{\tan x}{x}\to1, not 0, since tan⁡x≈x\tan x\approx x near 0. Hence only statement 1 is correct.
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Original question written in the NDA 2018 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake