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Saturday, 10 October

JEE Main Maths · Limits and Derivatives

L'Hopital's Rule in JEE Main

L'Hopital's Rule (Maths, Limits and Derivatives) came 24 times in 24 of the 42 JEE Main shifts Lumi analysed, across 8 distinct ideas. 15 of them came in 2024–26; it was last asked in 2026.

Questions
24
in 24 shifts
Ideas asked
8
5 asked more than once
Numerical answer
6
25% of its questions
Pattern
Asked most years
last asked 2026
Questions each year
0361’172’202’212’222’235’246’254’26

In the shifts Lumi analysed for each year.

Difficulty and format
Easy 2Medium 13Hard 9

Format

  • Single correct17 Q
  • Numerical value6 Q
  • Two statements1 Q
Ideas inside the topic
8 ideas
Where it sits

Part of Limits and Derivatives, which had 24 questions in all. L'Hopital's Rule accounts for 100% of them. In Lumi's JEE taxonomy it belongs to Calculus.

Some questions sit in other chapters too: Continuity and Differentiability.

Every question

All 24 L'Hopital's Rule questions

Newest first, each with its options and official answer.

  1. 2 Apr 2026, Shift 1 · Q16If lim⁡x→2sin⁡(x3−5x2+ax+b)(x−1−1)log⁡e(x−1)=m\lim_{x\to2}\frac{\sin\left(x^3-5x^2+ax+b\right)}{\left(\sqrt{x-1}-1\right)\log_e(x-1)}=m, then a+b+ma+b+m is equal to :HardSingle correct
  2. 5 Apr 2026, Shift 1 · Q17The product of all possible values of α\alpha, for which…HardSingle correct
  3. 5 Apr 2026, Shift 2 · Q17Let f(x)=lim⁡y→0(1−cos⁡(xy))tan⁡(xy)y3f(x)=\lim_{y\to 0}\frac{(1-\cos(xy))\tan(xy)}{y^3}. Then the number of solutions of the equation f(x)=sin⁡xf(x)=\sin x, x∈Rx \in \mathbf{R}…MediumSingle correct
  4. 6 Apr 2026, Shift 1 · Q17The value of lim⁡x→0(x2sin⁡2xx2−sin⁡2x)\lim_{x\to 0}\left(\frac{x^2\sin^2 x}{x^2-\sin^2 x}\right) is:MediumSingle correct
  5. 2 Apr 2025, Shift 1 · Q7For α,β,γ∈R\alpha, \beta, \gamma\in\mathbf{R}, if lim⁡x→0x2sin⁡αx+(γ−1)ex2sin⁡2x−βx=3\lim_{x\to0}\frac{x^2\sin\alpha x+(\gamma-1)e^{x^2}}{\sin2x-\beta x}=3, then…MediumSingle correct
  6. 2 Apr 2025, Shift 2 · Q17If lim⁡x→0cos⁡(2x)+acos⁡(4x)−bx4\lim_{x\to0} \frac{\cos(2x) + a\cos(4x) - b}{x^4} is finite, then (a+b)(a+b) is equal to :MediumSingle correct
  7. 3 Apr 2025, Shift 2 · Q25If lim⁡x→0(tan⁡xx)1x2=p\lim_{x \to 0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^2}} = p, then 96log⁡ep96 \log_e p is equal to ___________.MediumNumerical value
  8. 7 Apr 2025, Shift 1 · Q16…MediumSingle correct
  9. 7 Apr 2025, Shift 2 · Q23For t>−1t>-1, let αt\alpha_t and βt\beta_t be the roots of the equation…HardNumerical value
  10. 8 Apr 2025, Shift 2 · Q16Given below are two statements: Statement I :…MediumTwo statements
  11. 8 Apr 2024, Shift 1 · Q26The value of lim⁡x→02(1−cos⁡xcos⁡2xcos⁡3x3⋯cos⁡10x10x2)\lim_{x\to0}2\left(\frac{1-\cos x\sqrt{\cos2x}\sqrt[3]{\cos3x}\cdots\sqrt[10]{\cos10x}}{x^2}\right) is ____________.MediumNumerical value
  12. 9 Apr 2024, Shift 2 · Q5lim⁡x→0e−(1+2x)12xx\lim_{x\to0}\frac{e-(1+2x)^{\frac{1}{2x}}}{x} is equal toHardSingle correct
  13. 29 Jan 2024, Shift 1 · Q8…MediumSingle correct
  14. 31 Jan 2024, Shift 1 · Q6lim⁡x→0e2∣sin⁡x∣−2∣sin⁡x∣−1x2\lim_{x\to0}\frac{e^{2|\sin x|}-2|\sin x|-1}{x^2}MediumSingle correct
  15. 31 Jan 2024, Shift 2 · Q25If lim⁡x→0ax2ex−blog⁡e(1+x)+cxe−xx2sin⁡x=1\lim_{x\to0}\frac{ax^2e^x-b\log_e(1+x)+cxe^{-x}}{x^2\sin x}=1, then 16(a2+b2+c2)16(a^2+b^2+c^2) is equal to ________.HardNumerical value
  16. 8 Apr 2023, Shift 2 · Q9If α>β>0\alpha > \beta > 0 are the roots of the equation ax2+bx+1=0ax^2 + bx + 1 = 0, and…HardSingle correct
  17. 13 Apr 2023, Shift 2 · Q9If lim⁡x→0eax−cos⁡(bx)−cxe−cx21−cos⁡(2x)=17\lim_{x\to0}\dfrac{e^{ax}-\cos(bx)-\dfrac{cxe^{-cx}}{2}}{1-\cos(2x)}=17, then 5a2+b25a^2+b^2 is equal toHardSingle correct
  18. 25 Jul 2022, Shift 1 · Q5If lim⁡n→∞(n2−n−1+nα+β)=0\lim_{n\to\infty}\left(\sqrt{n^2-n-1}+n\alpha+\beta\right)=0, then 8(α+β)8(\alpha+\beta) is equal to :MediumSingle correct
  19. 28 Jul 2022, Shift 1 · Q29lim⁡x→0((x+2cos⁡x)3+2(x+2cos⁡x)2+3sin⁡(x+2cos⁡x)(x+2)3+2(x+2)2+3sin⁡(x+2))100x\lim_{x\to0}\left(\frac{(x+2\cos x)^3+2(x+2\cos x)^2+3\sin(x+2\cos x)}{(x+2)^3+2(x+2)^2+3\sin(x+2)}\right)^{\frac{100}{x}} is equal to…HardNumerical value
  20. 3 Aug 2021, Shift 2 · Q68lim⁡x→0(1+x)1ex−1\lim_{x\to0}(1+x)^{\frac{1}{e^x-1}} is equal toEasySingle correct
  21. 18 Mar 2021, Shift 1 · Q72If lim⁡x→0sin⁡−1x−tan⁡−1x3x3\lim_{x\to0}\dfrac{\sin^{-1}x-\tan^{-1}x}{3x^3} is equal to L, then the value of (6L+1)(6L+1) is :MediumSingle correct
  22. 8 Jan 2020, Shift 1 · Q57lim⁡x→0(3x2+27x2+2)1/x2\lim_{x\to 0}\left(\dfrac{3x^2+2}{7x^2+2}\right)^{1/x^2} is equal to :EasySingle correct
  23. 2 Sep 2020, Shift 1 · Q72If lim⁡x→1x+x2+x3+…+xn−nx−1=820, (n∈N)\lim_{x\to 1}\frac{x+x^{2}+x^{3}+\ldots+x^{n}-n}{x-1}=820,\ (n\in\mathbf{N}) then the value of n is equal to ___________.MediumNumerical value
  24. 2 Apr 2017 (offline) · Q70lim⁡x→π2cot⁡x−cos⁡x(π−2x)3\lim_{x\to\frac{\pi}{2}}\frac{\cot x-\cos x}{(\pi-2x)^3} equals :HardSingle correct