Skip to content
Miscellaneous 7 II · Q126

Q.lim⁡x→1[abx−axbx2−1]\displaystyle\lim_{x\to 1}\left[\frac{ab^x-a^xb}{x^2-1}\right]

Maharashtra MsbshseTextbookSubjectiveImportance★★★★★est
20% · 28/137 Questions
🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

abx−axb=ab(bx−1−ax−1)ab^x-a^xb=ab\left(b^{x-1}-a^{x-1}\right) (factoring out abab and using bx=b⋅bx−1b^x=b\cdot b^{x-1}, ax=a⋅ax−1a^x=a\cdot a^{x-1}). At x=1x=1, bx−1−ax−1=1−1=0b^{x-1}-a^{x-1}=1-1=0, matching the numerator's vanishing. Dividing by (x−1)(x-1): ab[bx−1−1x−1−ax−1−1x−1]→ab(log⁡b−log⁡a)ab\left[\dfrac{b^{x-1}-1}{x-1}-\dfrac{a^{x-1}-1}{x-1}\right]\to ab(\log b-\log a) as x→1x\to1 (substituting u=x−1→0u=x-1\to0 makes this the standard (cu−1)/u→log⁡c(c^u-1)/u\to\log c limit). Dividing further …

Start your 14-day free trial to unlock the full solution →

Unlock everything free for 14 days

  • Full step-by-step solutions
  • Concept-first explanations
  • Methods, shortcuts & mistakes
  • PYQ mapping + timed mock tests
Start 14-day free trial →See Plans & Pricing →

Full access for 14 days. No credit card required.

←Previous questionQ125 — limx→ 2[(log x-log 2)/(x-2)]→Next questionQ127 — limx→ 0[((5x-1)²)/((2x-1)log(1+x))]