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NDA 2023 Maths · Mathematics · Limits · medium

Lim x→0(3^x-2^x/(x) is equal to

Single correct option

lim⁡x→03x−2xx\displaystyle\lim_{x\to0}\dfrac{3^x-2^x}{x} is equal to
  1. A. ln⁡6\ln6
  2. B. ln⁡23\ln\dfrac23
  3. C. 11
  4. D. ln⁡32\ln\dfrac32
Show answer and solution

Answer: (D) ln⁡32\ln\dfrac32

Write 3x−2xx=3x−1x−2x−1x→ln⁡3−ln⁡2=ln⁡32\dfrac{3^x-2^x}{x}=\dfrac{3^x-1}{x}-\dfrac{2^x-1}{x}\to\ln3-\ln2=\ln\dfrac32, using lim⁡ax−1x=ln⁡a\lim\dfrac{a^x-1}{x}=\ln a. The value ln⁡6\ln6 results from adding instead of subtracting.
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