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Limit - JEE Main Previous Year Question with Solutions

Limit questions appear in JEE Main every year, typically 1–2 problems per paper. The most tested techniques are the Sandwich (Squeeze) theorem, standard trigonometric limits, L'Hôpital's rule, and the 1 − cos 2x identity. This page covers fully solved JEE Main previous year limit questions from 2010 to 2018 with step-by-step explanations.
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Limit - JEE Main Previous Year Question with Solutions

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JEE Main Previous Year Question of Math with Solutions are available at eSaral. Practicing JEE Main Previous Year Papers Questions of mathematics will help the JEE aspirants in realizing the question pattern as well as help in analyzing weak & strong areas. eSaral helps the students in clearing and understanding each topic in a better way. eSaral is providing complete chapter-wise notes of Class 11th and 12th both for all subjects. Besides this, eSaral also offers NCERT Solutions, Previous year questions for JEE Main and Advance, Practice questions, Test Series for JEE Main, JEE Advanced and NEET, Important questions of Physics, Chemistry, Math, and Biology and many more. Download eSaral app for free study material and video tutorials.

Q. Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a positive increasing function with $\lim _{x \rightarrow \infty} \frac{\mathrm{f}(3 \mathrm{x})}{\mathrm{f}(\mathrm{x})}=1 .$ Then $\lim _{\mathrm{x} \rightarrow \infty} \frac{\mathrm{f}(2 \mathrm{x})}{\mathrm{f}(\mathrm{x})}=$ (1) 1 (2) $\frac{2}{3}$ (3) $\frac{3}{2}$ (4) 3 [AIEEE-2010]
Ans. (1) $\mathrm{f}(\mathrm{x})$ is a positive increasing function $\Rightarrow 0<\mathrm{f}(\mathrm{x})<\mathrm{f}(2 \mathrm{x})<\mathrm{f}(3 \mathrm{x})$ $\Rightarrow 0<1<\frac{\mathrm{f}(2 \mathrm{x})}{\mathrm{f}(\mathrm{x})}<\frac{\mathrm{f}(3 \mathrm{x})}{\mathrm{f}(\mathrm{x})}$ $\Rightarrow \lim _{x \rightarrow \infty} 1 \leq \lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)} \leq \lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}$ By sandwich theorem. $\Rightarrow \lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}=1$
Q. $\lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos \{2(x-2)\}}}{x-2}\right)$ (1) equals $-\sqrt{2}$ (2) equals $\frac{1}{\sqrt{2}}$ (3) does not exist (4) equals $\sqrt{2}$ [AIEEE-2011]
Ans. (3) $\lim _{x \rightarrow 2} \frac{\sqrt{1-\cos 2(x-2)}}{(x-2)}=\lim _{x \rightarrow 2} \frac{\sqrt{2 \sin ^{2}(x-2)}}{(x-2)}$ $=\lim _{x \rightarrow 2} \frac{\sqrt{2}|\sin (x-2)|}{(x-2)}$ $\mathrm{RHL}$ at $\mathrm{x}=2, \lim _{\mathrm{h} \rightarrow 0} \frac{\sqrt{2}|\sin (2+\mathrm{h}-2)|}{(2+\mathrm{h})-2}=\lim _{\mathrm{h} \rightarrow 0} \frac{\sqrt{2}|\sinh |}{\mathrm{h}}$ $=\lim _{h \rightarrow 0} \frac{\sqrt{2} \sinh }{-h}=-\sqrt{2}$ LHL at $\mathrm{x}=2, \lim _{\mathrm{h} \rightarrow 0} \frac{\sqrt{2}|\sin (2-\mathrm{h}-2)|}{(2-\mathrm{h})-2}$ $=\lim _{h \rightarrow 0} \frac{\sqrt{2}|\sin (-h)|}{-h}=\lim _{h \rightarrow 0} \frac{\sqrt{2} \sinh }{-h}=-\sqrt{2}$ $\because \mathrm{LHL} \neq \mathrm{RHL}$ Hence, limit does not exist.
Q. Let $f: \mathrm{R} \rightarrow[0, \infty)$ be such that $\lim _{x \rightarrow 5} f(x)$ exists and $\lim _{x \rightarrow 5} \frac{(f(x))^{2}-9}{\sqrt{|x-5|}}=0 .$ Then $\lim _{x \rightarrow 5} \operatorname{Lim}_{x \rightarrow 5}(x)$ equal – (1) 3             (2) 0               (3) 1                  (4) 2 [AIEEE-2011]
Ans. (1) $\lim _{x \rightarrow 5} \frac{(f(x))^{2}-9}{\sqrt{|x-5|}}=0$ $\therefore \quad$ Question must be in $\frac{0}{0}$ form $\therefore \quad(f(5))^{2}-9=0$ $\Rightarrow \quad f(5)=3$
Q. $\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^{2} x\right)}{x^{2}}$ is equal to: (1) $\frac{\pi}{2}$ (2) 1 $(3)-\pi$ (4)$\pi$ [JEE Mains Offline-2014]
Ans. (4) $\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^{2} x\right)}{x^{2}}$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\sin \left(\pi-\pi \sin ^{2} x\right)}{x^{2}}$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\sin \left(\pi \sin ^{2} x\right)}{\pi \sin ^{2} x} \times \pi \frac{\sin ^{2} x}{x^{2}}$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\sin \left(\pi \sin ^{2} x\right)}{\pi \sin ^{2} x} \times \pi \times \lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{2}$ $\Rightarrow 1 \times \pi \times 1$ $=\pi$
Q. If $\lim _{x \rightarrow 2} \frac{\tan (x-2)\left\{x^{2}+(k-2) x-2 k\right\}}{x^{2}-4 x+4}=5$ then $k$ is equal to (1) 3                  (2) 1                  (3) 0                     (4) 2 [JEE Mains Online-2014]
Ans. (1) If $\lim _{x \rightarrow 2} \frac{\tan (x-2)\left[x^{2}+k x-2 k-2 x\right]}{(x-2)^{2}}=5$ $\lim _{x \rightarrow 2}\left(\frac{\tan (x-2)}{(x-2)}\right) \frac{(x+k)(x-2)}{(x-2)}=5$ 1. (2 + k) = 5 K = 3
Q. Let $\mathrm{p}=\lim _{x \rightarrow 0+}\left(1+\tan ^{2} \sqrt{\mathrm{x}}\right)^{\frac{1}{2 \mathrm{x}}}$ then log $\mathrm{p}$ is equal to :- (1) $\frac{1}{4}$ (2) 2 (3) 1 (4) $\frac{1}{2}$ [JEE Mains -2016]
Ans. (4) $\mathrm{p}=\mathrm{e}^{\mathrm{x} \rightarrow 0^{\frac{1}{2}}\left(\frac{\tan \sqrt{x}}{\sqrt{x}}\right)^{2}}=\sqrt{\mathrm{e}}$ $\log \mathrm{p}=\frac{1}{2}$
Q. $\lim _{n \rightarrow \infty}\left(\frac{(n+1)(n+2) \ldots .3 n}{n^{2 n}}\right)^{1 / n}$ is equal to :- (1) $3 \log 3-2$ (2) $\frac{18}{\mathrm{e}^{4}}$b (3) $\frac{27}{\mathrm{e}^{2}}$ (4) $\frac{9}{\mathrm{e}^{2}}$ [JEE Mains 2016]
Ans. (3)
Q. $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x-\cos x}{(\pi-2 x)^{3}}$ equals :- (1) $\frac{1}{4}$ (2) $\frac{1}{24}$ (3) $\frac{1}{16}$ (4) $\frac{1}{8}$ [JEE Mains -2017]
Ans. (3) $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot x(1-\sin x)}{-8\left(x-\frac{\pi}{2}\right)^{3}}$ $=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan \left(\frac{\pi}{2}-x\right)}{8\left(\frac{\pi}{2}-x\right)} \frac{\left(1-\cos \left(\frac{\pi}{2}-x\right)\right)}{\left(\frac{\pi}{2}-x\right)^{2}}$ $=\frac{1}{8} \cdot 1 \cdot \frac{1}{2}=\frac{1}{16}$
Q. For each $\mathrm{t} \in \mathrm{R},$ let $[\mathrm{t}]$ be the greatest integer less than or equal to t. Then $\lim _{\mathrm{x} \rightarrow 0+} \mathrm{x}\left(\left[\frac{1}{\mathrm{x}}\right]+\left[\frac{2}{\mathrm{x}}\right]+\ldots \ldots+\left[\frac{15}{\mathrm{x}}\right]\right)$\ (1) is equal to 15. (2) is equal to 120. (3) does not exist (in R). (4) is equal to 0. [JEE Mains -2018]
Ans. (2)
Q. $\lim _{x \rightarrow 0} \frac{(27+x)^{1 / 3}-3}{9-(27+x)^{2 / 3}}$ equals : $(1)-\frac{1}{3}$ (2) $\frac{1}{6}$ $(3)-\frac{1}{6}$ (4) $\frac{1}{3}$ [JEE Mains -2018]
Ans. (3)
Q. $\lim _{x \rightarrow 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^{2}}$ equals :- $(1)-\frac{1}{2}$ (2) $\frac{1}{4}$ (3) $\frac{1}{2}$ (4) 1 [JEE Mains -2018]
Ans. (3)

Frequently Asked Questions

Find answers to common questions.

Is L'Hôpital's rule sufficient for all JEE Main limit problems?

No. L'Hôpital's rule applies only to 0/0 or ∞/∞ indeterminate forms, and repeated differentiation can be slower than factorisation or standard substitution. Several JEE Main questions — particularly those involving the greatest integer function or the Sandwich theorem — cannot be solved using L'Hôpital's rule at all. Build fluency with multiple methods.

Which topics in limits are most important for JEE Main?

The highest-frequency topics are: (1) Sandwich theorem, (2) standard trigonometric limits using sin x/x and tan x/x forms, (3) exponential limits of the 1^∞ type, (4) limits involving the greatest integer function, and (5) limits of sequences expressed as Riemann sums. These five areas cover roughly 85% of JEE Main limit questions from 2010 to 2024.

How many limit questions come in JEE Main each year?

JEE Main typically includes 1–2 questions from limits per session (the exam is held in multiple sessions per year). The chapter "Limits, Continuity and Differentiability" as a whole contributes 2–3 questions. Based on the official NTA syllabus and analysis of 2019–2024 papers, limits problems appear in roughly 70% of all JEE Main sessions.

Where can I find more JEE Main previous year questions for maths?

eSaral provides chapter-wise JEE Main previous year questions for the entire Class 11 and 12 mathematics syllabus. For the underlying theory and worked examples, start with the NCERT Solutions for Class 11 Maths and NCERT Solutions for Class 12 Maths. The eSaral app also contains video solutions for every question on this page, taught by Kota-quality IIT Bombay faculty.

What is the Sandwich theorem and when is it used in JEE Main?

The Sandwich (Squeeze) theorem states: if g(x) ≤ f(x) ≤ h(x) near a point and lim g(x) = lim h(x) = L, then lim f(x) = L. It is used in JEE Main when direct substitution fails and the function can be bounded above and below by simpler functions whose limits are equal — the 2010 AIEEE question (Q1 above) is a textbook example.

How should a JEE dropper revise limits efficiently?

Solve previous year questions topic-by-topic first (2010–2019), then shift to full mixed-paper practice for 2020–2024. Identify the two or three technique types you get wrong most often and drill only those. eSaral's 5-layer doubt-solving system — taught by IIT Bombay faculty — ensures you get a concept-level explanation, not just an answer, when you are stuck.

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Comments

Krishna Mali
Aug. 31, 2025, 6:35 a.m.
Good solution
Yugesh Kumar
Sept. 9, 2025, 6:35 a.m.
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Kide diya
April 22, 2025, 6:35 a.m.
Pyq on limit last three year's
Kide diya
April 22, 2025, 6:35 a.m.
Pyq on limit last three year's
Abheeshna Dey
Nov. 15, 2024, 6:31 p.m.
More varieties of questions from last 10 years of JEE Mains&Advanced should be provided.
Sharath
May 12, 2024, 6:35 a.m.
Question no.4 is wrong
K. Lalith Aditya
Oct. 10, 2024, 6:35 a.m.
Yes
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Nov. 29, 2023, 7:05 a.m.
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Sept. 29, 2023, 10 p.m.
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Komal
Sept. 20, 2023, 6:35 a.m.
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Komal
Sept. 20, 2023, 6:35 a.m.
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March 9, 2023, 6:49 p.m.
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Aug. 14, 2021, 2:23 p.m.
to easy
Aarzoo
Aug. 9, 2021, 11:49 p.m.
Nice questions🙋
Priya
May 2, 2021, 12:20 p.m.
Level of questions are easy
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April 29, 2021, 6:01 p.m.
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Ayush
April 15, 2021, 5:56 p.m.
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D rugved
April 12, 2021, 7:20 p.m.
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ULL THE BULL
March 8, 2021, 5:22 p.m.
Some questions dont have correct solutions
Rohini
Feb. 21, 2021, 11:19 a.m.
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Feb. 20, 2021, 11:02 a.m.
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Irene
Feb. 14, 2021, 10:22 a.m.
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Rohith
Dec. 8, 2020, 11:51 a.m.
Hey! Thanks for the questions.But there are no solutions for some questions please look into those question kindly update answers it would be helpful Thank you.
Manoj
Aug. 31, 2020, 7:57 p.m.
Questions are overlapped we can't understand
Sam
Aug. 27, 2020, 2:20 p.m.
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Ali
Aug. 25, 2020, 2:54 p.m.
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Aug. 16, 2020, 9:08 p.m.
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Aug. 16, 2020, 9:07 p.m.
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Aug. 16, 2020, 9:06 p.m.
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Aug. 16, 2020, 9:06 p.m.
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Priyanka
Aug. 15, 2020, 12:33 p.m.
Thanks
alphadk
Aug. 1, 2020, 12:22 p.m.
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Saptarshi Mukherjee
July 20, 2020, 12:18 p.m.
Good collection of problems but very bad printing of questions.........you should look after it!!!!
B. Shivanireddy1
July 9, 2020, 2 p.m.
thank you , but if it is pdf it will be more clearity
Ayush
July 2, 2020, 7:26 a.m.
Thanks for help
Akhil
May 22, 2020, 9:49 a.m.
Chill bro
A
April 29, 2020, 4:03 p.m.
Why problem in question upon question boss
Ran
March 25, 2020, 11:08 p.m.
No answer description
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