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NDA 2024 Maths · Mathematics · Limits · medium

Limits is/are correct? 1. lim x→0(|x|)/(x)=1. 2. lim x→01/x exists as a finite real number. Select the correct answer using the code given below.

Single correct option

Which of the following statements about limits is/are correct?
1. lim⁡x→0∣x∣x=1\displaystyle\lim_{x\to0}\frac{|x|}{x}=1.
2. lim⁡x→01x\displaystyle\lim_{x\to0}\frac1x exists as a finite real number.
Select the correct answer using the code given below.
  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: (D) Neither 1 nor 2

For x>0x>0, ∣x∣x=1\frac{|x|}{x}=1, but for x<0x<0 it equals −1-1; the one-sided limits differ, so the limit does not exist and statement 1 is false. As x→0x\to0, 1x\frac1x grows without bound with sign depending on the side, so no finite limit exists and statement 2 is false. Trap: evaluating only from the right.
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Original question written in the NDA 2024 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