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Sunday, 25 October

JEE Main · Maths · Class 11

Limits and Derivatives: JEE Main previous year questions

64 questions in 61 of the 108 JEE Main shifts Lumi analysed, asked in 8 of 8 years. Here is what this chapter tests and how.

Study this chapter
Questions
64
in 61 of 108 shifts · 8 of 8 years
Average a shift
0.6
of 25 Maths · 0.6 since 2024 0.0
Typical range
0.4–0.8
questions a shift; fewest 0.3, most 0.9
Weighted estimate
0.5
model, recent years count more · about 2.1 marks
  • Typical range (fact)
  • Average a shift (fact)
  • Weighted estimate (model)

Questions are counted per shift, out of the 25 each subject has, in the shifts Lumi analysed; papers that offered 30 a subject (2017, 2021–2024) are scaled to 25. Typical range: the middle six of the eight years (2017, 2020–2026), leaving out the single highest and lowest year. The average and the range are historical fact. The weighted estimate is a model: each year's share, with recent years counting more (half-life three years), and chapters NTA dropped from the syllabus set near zero. It is a guide to effort, not a forecast of the paper.

What to do

How to spend time on Limits and Derivatives

Priority

Lower priority

One of the 7 Maths chapters that shared the last fifth of Maths questions in the 108 shifts analysed. How bands are set

Past questions
64in 61 of 108 shifts
Study plan gives it
11 hover 12 months; 9 h over 6
Study this chapter →

Ideas asked most often

  1. Simplifying with Standard Limits Before L'Hopital13×
  2. Choosing Series Expansion over L'Hopital10×
  3. Determining Unknown Constants from Finite Limits6×
  4. e-Power Shortcut for 1^infinity Forms6×
  5. Indeterminate Power Forms5×

41 of the 64 ideas in Lumi’s map of this chapter have not been asked or touched in the shifts analysed. Of its idea-tagged questions from 2024 on, 0% tested an idea not asked in an earlier year, so learn the whole chapter, not just the list above.

Within each subject, chapters in NTA's 2026 syllabus are ranked by their average questions a shift over all 8 years analysed (history, not the model). Going down that list, the chapters that together held the first half of the subject's questions are Must do; the next ones, up to 80%, are Should do; the rest are Lower priority.

Key findings

Limits and Derivatives at a glance

  1. 01

    100%

    of its 21 questions from 2024 on that Lumi matched to an idea tested one JEE Main had asked in an earlier year; 100% came from a topic asked in an earlier year.

  2. 02

    64%

    of the 64 ideas in Lumi's map of the chapter have not been asked or touched on in the shifts analysed.

  3. 03

    22%

    of its questions asked for a numerical answer, 0.7 times the Maths average.

  4. 04

    27%

    of its questions were rated hard, against 26% across all Maths questions in the same shifts.

Questions a shift, year by year

Out of the 25 Maths questions in a shift. Years differ in how many shifts Lumi analysed, so each year is scaled to one shift.

0240.9’170.6’200.6’210.4’220.4’230.6’240.8’250.3’26

Raw count: 2017: 1 in 1 shift · 2020: 7 in 12 shifts · 2021: 16 in 24 shifts · 2022: 12 in 24 shifts · 2023: 6 in 12 shifts · 2024: 13 in 18 shifts · 2025: 6 in 8 shifts · 2026: 3 in 9 shifts.

Kind of question

SHARE OF QUESTIONS
  • Calculation42%
  • Multi-step25%
  • Numerical answer22%
  • Application6%
  • Two statements3%
  • Conceptual2%

Most common asks: unknown constants from finite limit (7); limit of one-to-infinity form (4); parameter values from trigonometric limit (1).

Profile

This chapter in numbers

Where it ranks, how steady it is, and what it costs to study.

Rank, all JEE chapters
#65
of 70 by weighted estimate
Rank in Maths
#20
of 21 chapters
Share of Maths
2.3%
of its questions, all shifts analysed
Busiest year
2017
0.9 questions a shift
Year to year
Some swing
varies by about 0.2 questions a shift
Long-run direction
-0.03
questions a shift, per year
Easy questions
5%
27% rated hard
Numerical answer
22%
0.7× the Maths rate
With a figure
0%
graph, diagram or structure
Study time
11 h
9 h on a 6-month plan
Marks per study hour
0.18
#68 of 70 chapters
Foundation for others
56%
of its ideas are used in other chapters

Difficulty, year by year

Each question rated easy, medium or hard. Each bar is that year’s split, so years compare directly; the number is the questions in that year’s analysed shifts (1 to 24 shifts a year).

  • 20171
  • 20207
  • 202116
  • 202212
  • 20236
  • 202413
  • 20256
  • 20263
  • Easy
  • Medium
  • Hard

What it asks you to do

SHARE OF QUESTIONS
  • Calculation50%
  • Multi-step41%
  • Application8%
  • Conceptual2%

Coverage

How much of the chapter JEE Main has used

We split the chapter into 64 ideas. In the 108 shifts analysed, JEE Main built a question around 18 of them. Most of the chapter has never been the main idea of a question, so past questions alone leave most of it out.

Asked as the main idea
18
Touched on in a question
5
Never asked
41

Topics

What repeats, topic by topic

This chapter's questions per topic in each year's analysed shifts. The number is the count; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. 2 questions had no close topic in Lumi's map and count for the chapter only.

’17’20’21’22’23’24’25’26
014+questions in that year’s analysed shifts, shaded per shift
  • L'Hopital's Rule

    As a topic: 61 questions in 8 of 8 years, most often single correct, 25% numerical answer.

  • Continuity and Differentiability

    As a topic: 94 questions in 7 of 8 years, most often single correct, 31% numerical answer. Recent asks: statement I/II on differentiability and monotonicity; find constant from continuity then integral; count discontinuities of composite piecewise function.

  • Integral as Limit of Sum

    As a topic: 13 questions in 4 of 8 years, most often single correct, 15% numerical answer.

  • AP and GP

    As a topic: 136 questions in 8 of 8 years, most often single correct, 32% numerical answer. Recent asks: sum of series then solve for angle; combined AP and GP parameter problem; numerical: GP sum from trig extrema.

Drill down

Topic, idea, question

Open a topic to see every idea JEE Main tested inside it, then open an idea to read the questions. Each row shows its 8 years.

  • Open55 · 86%7141031163

    16 ideas inside

    • OpenRepeated13 · 24%222322
    • OpenRepeated10 · 18%222
    • OpenRepeated5 · 9%22
    • OpenRepeated5 · 9%23
    • OpenRepeated5 · 9%3
    • OpenRepeated4 · 7%2
    • OpenRepeated3 · 5%
    • OpenRepeated2 · 4%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
  • Open2 · 3%2
  • Open2 · 3%
  • Open1 · 2%
  • Open1 · 2%
  • Open1 · 2%
  • 6 Apr ’24 S2 · Q9
    lim⁡n→∞(12−1)(n−1)+(22−2)(n−2)+⋯+((n−1)2−(n−1))⋅1(13+23+⋯+n3)−(12+22+⋯+n2)\lim_{n\to\infty}\frac{(1^2-1)(n-1)+(2^2-2)(n-2)+\cdots+((n-1)^2-(n-1))\cdot 1}{(1^3+2^3+\cdots+n^3)-(1^2+2^2+\cdots+n^2)} is equal to :

    mediumSingle correct

  • 6 Apr ’23 S2 · Q9
    …

    hardSingle correct

Squares are the 8 years with analysed shifts, 2017 to 2026. A number is the questions that year when there is more than one; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. Share is of the level above.

Ideas

The exact ideas JEE Main has asked

Inside the topics, 17 different ideas were tested; 8 came more than once. These are the repeats, then the ideas new since 2024.

Dots are the 8 years with analysed shifts, 2017 to 2026.

Every question

All 64 Limits and Derivatives questions

Newest first, shift by shift. The newest 40 show their opening words; older ones name the idea they tested. Each opens with its options and the official answer.

JEE Main 20263 questions

  1. 5 Apr 2026, Shift 1 · Q17The product of all possible values of α\alpha, for which…HardSingle correct
  2. 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
  3. 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

JEE Main 20256 questions

  1. 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
  2. 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
  3. 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
  4. 7 Apr 2025, Shift 1 · Q16…MediumSingle correct
  5. 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
  6. 8 Apr 2025, Shift 2 · Q16Given below are two statements: Statement I :…MediumTwo statements

JEE Main 202413 questions

  1. 4 Apr 2024, Shift 1 · Q24If lim⁡x→1(5x+1)1/3−(x+5)1/3(2x+3)1/2−(x+4)1/2=m5n(2n)2/3\lim_{x\to1}\frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}=\frac{m\sqrt{5}}{n(2n)^{2/3}}, where gcd⁡(m,n)=1\gcd(m, n)=1, then…HardNumerical value
  2. 4 Apr 2024, Shift 2 · Q9Let f(x)=∫0x(t+sin⁡(1−et))dt, x∈Rf(x)=\int_0^x\left(t+\sin\left(1-e^t\right)\right)dt,\ x\in\mathbb{R}. Then, lim⁡x→0f(x)x3\lim_{x\to0}\frac{f(x)}{x^3} is equal toHardSingle correct
  3. 5 Apr 2024, Shift 2 · Q23Let a>0a > 0 be a root of the equation 2x2+x−2=02x^2+x-2=0. If…HardNumerical value
  4. 6 Apr 2024, Shift 2 · Q9lim⁡n→∞(12−1)(n−1)+(22−2)(n−2)+⋯+((n−1)2−(n−1))⋅1(13+23+⋯+n3)−(12+22+⋯+n2)\lim_{n\to\infty}\frac{(1^2-1)(n-1)+(2^2-2)(n-2)+\cdots+((n-1)^2-(n-1))\cdot 1}{(1^3+2^3+\cdots+n^3)-(1^2+2^2+\cdots+n^2)} is equal to :MediumSingle correct
  5. 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
  6. 8 Apr 2024, Shift 2 · Q25If α=lim⁡x→0+(etan⁡x−extan⁡x−x)\alpha = \lim_{x \to 0^+} \left(\frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}}\right) and…HardNumerical value
  7. 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
  8. 1 Feb 2024, Shift 1 · Q26Let {x}\{x\} denote the fractional part of xx and f(x)=cos⁡−1(1−{x}2)sin⁡−1(1−{x}){x}−{x}3f(x)=\frac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, x≠0x\neq0. If L and…HardNumerical value
  9. 1 Feb 2024, Shift 2 · Q9Let f(x)={x−1,x is even,2x,x is odd,f(x)=\begin{cases}x-1, & x \text{ is even,}\\ 2x, & x \text{ is odd,}\end{cases} x∈Nx\in\mathbf{N}. If for some a∈Na\in\mathbf{N},…MediumSingle correct
  10. 27 Jan 2024, Shift 1 · Q9If a=lim⁡x→01+1+x4−2x4a=\lim_{x\to0}\dfrac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4} and b=lim⁡x→0sin⁡2x2−1+cos⁡xb=\lim_{x\to0}\dfrac{\sin^2 x}{\sqrt{2}-\sqrt{1+\cos x}}, then the…MediumSingle correct
  11. 29 Jan 2024, Shift 1 · Q8…MediumSingle correct
  12. 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
  13. 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

JEE Main 20236 questions

  1. 6 Apr 2023, Shift 2 · Q9…HardSingle correct
  2. 8 Apr 2023, Shift 1 · Q10lim⁡x→0((1−cos⁡2(3x)cos⁡3(4x))(sin⁡3(4x)(log⁡e(2x+1))5))\lim_{x\to0}\left(\left(\frac{1-\cos^2(3x)}{\cos^3(4x)}\right)\left(\frac{\sin^3(4x)}{\left(\log_e(2x+1)\right)^5}\right)\right) is…MediumSingle correct
  3. 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
  4. 13 Apr 2023, Shift 1 · Q7Among (S1) : lim⁡n→∞1n2(2+4+6+……+2n)=1\lim_{n\to\infty}\frac{1}{n^2}(2+4+6+\ldots\ldots+2n)=1 (S2) :…MediumTwo statements
  5. 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
  6. 13 Apr 2023, Shift 2 · Q25Let f(x)=∑k=110k xkf(x)=\sum_{k=1}^{10}k\,x^k, x∈Rx\in\mathbb{R}. If 2f(2)+f′(2)=119(2)n+12f(2)+f'(2)=119(2)^n+1 then nn is equal to ______HardNumerical value

JEE Main 202212 questions

  1. 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
  2. 25 Jul 2022, Shift 2 · Q6lim⁡x→π482−(cos⁡x+sin⁡x)72−2sin⁡2x\lim_{x\to\frac{\pi}{4}}\frac{8\sqrt{2}-(\cos x+\sin x)^{7}}{\sqrt{2}-\sqrt{2}\sin 2x} is equal toMediumSingle correct
  3. 26 Jul 2022, Shift 2 · Q7Let β=lim⁡x→0αx−(e3x−1)αx(e3x−1)\beta=\lim_{x\to 0}\frac{\alpha x-(e^{3x}-1)}{\alpha x(e^{3x}-1)} for some α∈R\alpha\in\mathbb{R}. Then the value of α+β\alpha+\beta…MediumSingle correct
  4. 29 Jul 2022, Shift 1 · Q5If lim⁡x→0αex+βe−x+γsin⁡xxsin⁡2x=23\lim_{x \to 0} \frac{\alpha e^x + \beta e^{-x} + \gamma \sin x}{x \sin^2 x} = \frac{2}{3}, where…MediumSingle correct
  5. 25 Jun 2022, Shift 1 · Q9Let f(x)f(x) be a polynomial function such that f(x)+f′(x)+f′′(x)=x5+64f(x)+f'(x)+f''(x)=x^5+64. Then, the value of lim⁡x→1f(x)x−1\lim_{x\to1}\frac{f(x)}{x-1} is equal to :HardSingle correct
  6. 25 Jun 2022, Shift 2 · Q5…HardSingle correct
  7. 26 Jun 2022, Shift 1 · Q6lim⁡x→12sin⁡(cos⁡−1x)−x1−tan⁡(cos⁡−1x)\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\sin(\cos^{-1}x) - x}{1 - \tan(\cos^{-1}x)} is equal to :MediumSingle correct
  8. 26 Jun 2022, Shift 2 · Q4lim⁡x→0cos⁡(sin⁡x)−cos⁡xx4\lim_{x \to 0} \frac{\cos(\sin x) - \cos x}{x^4} is equal to :MediumSingle correct
  9. 27 Jun 2022, Shift 1 · Q5Let a be an integer such that lim⁡x→718−[1−x][x−3a]\lim_{x \to 7} \frac{18-[1-x]}{[x-3a]} exists, where [t][t] is greatest integer ⩽t\leqslant t. Then a is…MediumSingle correct
  10. 27 Jun 2022, Shift 2 · Q25Let [t][t] denote the greatest integer ≤t\leq t and {t}\{t\} denote the fractional part of tt. The integral value of α\alpha for which the…MediumNumerical value
  11. 28 Jun 2022, Shift 2 · Q24If lim⁡x→1sin⁡(3x2−4x+1)−x2+12x3−7x2+ax+b=−2\lim_{x\to1}\frac{\sin(3x^2-4x+1)-x^2+1}{2x^3-7x^2+ax+b}=-2, then the value of (a−b)(a-b) is equal to __________.MediumNumerical value
  12. 29 Jun 2022, Shift 2 · Q5The value of lim⁡x→1(x2−1)sin⁡2(πx)x4−2x3+2x−1\lim_{x \to 1} \frac{(x^2 - 1)\sin^2(\pi x)}{x^4 - 2x^3 + 2x - 1} is equal to:MediumSingle correct

JEE Main 202116 questions

  1. 3 Aug 2021, Shift 1 · Q65Indeterminate Power Forms · Hard · Single correct
  2. 3 Aug 2021, Shift 2 · Q68Indeterminate Power Forms · Easy · Single correct
  3. 4 Aug 2021, Shift 1 · Q67e-Power Shortcut for 1^infinity Forms · Medium · Single correct
  4. 4 Aug 2021, Shift 2 · Q66e-Power Shortcut for 1^infinity Forms · Medium · Single correct
  5. 26 Aug 2021, Shift 2 · Q65Method of Differences for Telescoping Series · Medium · Single correct
  6. 27 Aug 2021, Shift 1 · Q65Choosing Series Expansion over L'Hopital · Medium · Single correct
  7. 27 Aug 2021, Shift 2 · Q66Conjugation vs L'Hopital for Radicals · Medium · Single correct
  8. 31 Aug 2021, Shift 1 · Q67Simplifying with Standard Limits Before L'Hopital · Medium · Single correct
  9. 31 Aug 2021, Shift 2 · Q67Indeterminate Power Forms · Medium · Single correct
  10. 25 Feb 2021, Shift 1 · Q66e-Power Shortcut for 1^infinity Forms · Medium · Single correct
  11. 25 Feb 2021, Shift 2 · Q84Determining Unknown Constants from Finite Limits · Medium · Numerical value
  12. 26 Feb 2021, Shift 1 · Q67Trigonometric Pre-Simplification · Medium · Single correct
  13. 17 Mar 2021, Shift 1 · Q70Handling Greatest Integer Functions · Medium · Single correct
  14. 17 Mar 2021, Shift 2 · Q65Simplifying with Standard Limits Before L'Hopital · Medium · Single correct
  15. 17 Mar 2021, Shift 2 · Q70Greatest Integer Function Sums · Medium · Single correct
  16. 18 Mar 2021, Shift 1 · Q72Choosing Series Expansion over L'Hopital · Medium · Single correct

JEE Main 20207 questions

  1. 8 Jan 2020, Shift 1 · Q57e-Power Shortcut for 1^infinity Forms · Easy · Single correct
  2. 2 Sep 2020, Shift 1 · Q72First Principle Derivative Recognition · Medium · Numerical value
  3. 2 Sep 2020, Shift 2 · Q59e-Power Shortcut for 1^infinity Forms · Easy · Single correct
  4. 3 Sep 2020, Shift 1 · Q58Handling Greatest Integer Functions · Medium · Single correct
  5. 3 Sep 2020, Shift 1 · Q74Choosing Series Expansion over L'Hopital · Medium · Numerical value
  6. 3 Sep 2020, Shift 2 · Q57Fractional Power Substitution · Medium · Single correct
  7. 5 Sep 2020, Shift 1 · Q58Trigonometric Simplification Before Differentiation · Medium · Single correct

JEE Main 20171 question

  1. 2 Apr 2017 (offline) · Q70Repeated L'Hopital Application · Hard · Single correct

Questions from the official JEE Main papers published by NTA. Chapter, topic and idea tags, counts and notes are Lumi’s. Pattern: Asked most years.

More Maths, Class 11

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