PrepWiser › Learn › Mathematics › LimitsLimits — JEE Main PYQs & Practice Questions Limits is a Mathematics chapter in the JEE Main syllabus. This page brings together 110 previous-year and practice questions on Limits, organised into 5 topics: Concept of Limit, Standard Limits, L'Hopital's Rule, Sandwich Theorem and Indeterminate Forms. Most questions come with a step-by-step solution you can work through free in PrepWiser.
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Chapter weightage : how many questions each chapter had per shift in the 2024–26 papers.
Concept of Limit Let f(x)=lim _(n → ∞) Σᵣ₌₀ⁿ((tan (x / 2ʳ⁺¹)+tan ³(x / 2ʳ⁺¹))/(1-tan ²(x / 2ʳ⁺¹))) Then… If lim _(x → 1) (x + x² + x³ + ... + xⁿ - n)/(x - 1) = 820, (n ∈ N) then the value of n… Let a be an integer such that lim _(x → 7) (18 - [1 - x])/[x - 3a] exists, where [t] is… Let f(x)={x-1, x is even,; 2 x, x is odd, x ∈ N. If for some a ∈ N, f(f(f(a)))=21, then… Let [t] denote the greatest integer ≤ t and t denote the fractional part of t. The… If the function f(x)=(sin 3 x+α sin x-β cos 3 x)/x³, x ∈ R, is continuous at x=0, then… lim _(n → ∞) [ 1/n + n/(n + 1)² + n/(n + 2)² + ........ + n/(2n + 1)² ] is equal to : The value of lim _(n → ∞) (1 + 2 - 3 + 4 + 5 - 6 + ..... + (3n - 2) + (3n - 1) -… The set of all values of a for which lim _(x → a) ([x - 5] - [2x + 2]) = 0, where [α]… Let f : R → R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of lim _(x… If lim _(x → 1) (sin (3x² - 4x + 1) - x² + 1)/(2x³ - 7x² + ax + b) = - 2, then the value… Let [t] be the greatest integer less than or equal to t. Then the least value of p ∈ N… Let f, g and h be the real valued functions defined on ℝ as f(x)={x/|x|,, x ≠ 0; 1,, x=0… Let [t] denote the greatest integer ≤ t. If for some λ ∈ R - 1, 0, lim _(x → 0) | (1 - x… lim n → ∞{(2^(1/2)-2^(1/3))(2^(1/2)-2^(1/5)) … . .(2^(1/2)-2^(1/(2 n+1)))} is equal to : Standard Limits The value of lim _(x → 0 +) (cos ⁻¹(x - [x]²).sin ⁻¹(x - [x]²))/(x - x³), where [ x ]… If lim x → 0 (a x² eˣ-b log ₑ(1+x)+c x e⁻ˣ)/(x² sin x)=1, then 16(a²+b²+c²) is equal to… The value of lim _(h → 0) 2{ (√3 sin ( π/6 + h ) - cos ( π/6 + h ))/(√3 h( √3 cosh -… lim _(n → ∞) ( 1 + (1 + 1/2 + ........ + 1/n)/n² )ⁿ is equal to : Let a>0 be a root of the equation 2 x²+x-2=0. If lim x → 1/a 16(1-cos (2+x-2 x²))/(1-a… If lim _(x → 0) (sin ⁻¹x - tan ⁻¹x)/3x³ is equal to L, then the value of (6L + 1) is The value of lim x → 0 2((1-cos x √(cos 2 x) ∛(cos 3 x) … … . ¹⁰√(cos 10 x))/x²) is… If α, β are the distinct roots of x² + bx + c = 0, then lim _(x → β) (e^(2(x² + bx + c))… lim x → 0 (e-(1+2 x)^(1/(2 x)))/x is equal to The value of lim _(x → 1) ((x² - 1)sin ²(π x))/(x⁴ - 2x³ + 2x - 1) is equal to: If lim x → 1⁺ ((x-1)(6+λ cos (x-1))+μ sin (1-x))/(x-1)³=-1, where λ, μ ∈ ℝ, then λ+μ is… lim _(x → 0) ( tan ( π/4 + x ) )^(1/x) is equal to : lim _(x → 0) (sin ²( π cos ⁴x ))/x⁴ is equal to : The lowest integer which is greater than ( 1 + 1/10¹⁰⁰ )^(10¹⁰⁰) is ______________. For some a, b, let f(x)=|a+(sin x)/x, 1, b; a, 1+(sin x)/x, b; a, 1, b+(sin x)/x|, x ≠… If α > β > 0 are the roots of the equation a x²+b x+1=0, and lim x → 1/α((1-cos (x²+b… Let f:(-∞, ∞)-{0} → ℝ be a differentiable function such that f′(1)=lim a → ∞ a² f(1/a)… If for p ≠ q ≠ 0, the function f(x)=(⁷√p(729+x)-3)/(∛(729+q x)-9) is continuous at x=0… The value of limlogₑ(sec (ex)· sec (e²x)· … · sec (e¹⁰x))/(e²-e²ᶜᵒˢˣ) is equal to Let f(x)=lim((1-cos (xy))tan (xy))/y³. Then the number of solutions of the equation… L'Hopital's Rule lim _(x → π/2) ( 1/( x - π/2 )²∫_(x³)^(( π/2 )³) cos ( t^(1/3) )dt ) is equal to Let f:R→ (0,∞) be a twice differentiable function such that f(3)=18,f^'(3)=0 and… lim x → 0 48/x⁴ ∫0ˣ t³/(t⁶+1) d t is equal to ___________. Let f : R → R be a continuous function. Then lim _(x → π/4) (π/4∫₂^(sec ²x) f(x) dx)/(x²… Let f(x)=√(lim r → x{(2 r²[(f(r))²-f(x) f(r)])/(r²-x²)-r³ e^(f(r)/r)}) be differentiable… Let f(x) = x⁶ + 2x⁴ + x³ + 2x + 3, x ∈ R. Then the natural number n for which lim _(x →… Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If ∫₀ˣ… Let f(x)=∫₀ˣ(t+sin (1-eᵗ)) d t, x ∈ ℝ. Then, lim x → 0 f(x)/x³ is equal to lim _(x → 1/√2) (sin (cos ⁻¹x) - x)/(1 - tan (cos ⁻¹x)) is equal to : Let the slope of the line 45 x+5 y+3=0 be 27 r₁+(9 r₂)/2 for some r₁, r₂ ∈ ℝ. Then lim x… If lim x → 0 (3+α sin x+β cos x+log ₑ(1-x))/(3 tan ² x)=1/3 , then 2 α-β is equal to : If lim _(x → 0) (aeˣ - b cos x + ce⁻ˣ)/(x sin x) = 2, then a + b + c is equal to… If lim x → 1 ((5 x+1)^(1 / 3)-(x+5)^(1 / 3))/((2 x+3)^(1 / 2)-(x+4)^(1 / 2))=(m √5)/n(2… Let f:[-π/2, π/2] → R be a differentiable function such that f(0)=1/2. If the lim x → 0… Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then lim _(x… lim x → π/2((∫_(x³)^((π / 2)³)(sin (2 t^(1 / 3))+cos (t^(1 / 3))) d t)/(x-π/2)²) is… Sandwich Theorem Indeterminate Forms lim x → 0(((1-cos ²(3 x)/(cos ³(4 x)))((sin ³(4 x))/(log ₑ(2 x+1))⁵)) is equal to… Let x=2 be a root of the equation x²+px+q=0 and f(x) = { (1 - cos (x² - 4px + q² + 8q +… If α = lim _(x → π/4) (tan ³x - tan x)/(cos ( x + π/4 )) and β = lim _(x → 0) (cos… lim _(x → 0) (∫₀^(x²) ( sin √t )dt)/x³ is equal to : Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If lim… Let β=lim _(x → 0) (α x-(e³ˣ-1))/(α x(e³ˣ-1)) for some α ∈ ℝ. Then the value of α+β is : If lim x → ∞((e/(1-e))(1/e-x/(1+x)))ˣ=α, then the value of (log ₑ α)/(1+log ₑ α) equals : If lim _(x → ∞) ( √(x² - x + 1) - ax ) = b, then the ordered pair (a, b) is : Suppose lim _(x → 0) F(x)/x³ exists and is equal to L, where F(x) = | a + sin x/2, - b… If lim _(n → ∞) ( √(n² - n - 1) + nα + β ) = 0, then 8(α+β) is equal to : If the value of lim _(x → 0) (2 - cos x√(cos 2x) )^(( (x + 2)/x² )) is equal to eᵃ, then… If the function f(x)=(eˣ(eᵗᵃⁿˣ⁻ˣ-1)+log ₑ(sec x+tan x)-x)/(tan x-x) is continuous at… Let a₁, a₂, a₃, …, aₙ be n positive consecutive terms of an arithmetic progression. If d… The value of the limit lim _(θ → 0) (tan (π cos ²θ ))/(sin (2π sin ²θ )) is equal to : If lim _(x → 0) ((tan x)/x)^(1/x²)=p , then 96 log ₑ p is equal to____________ If lim _(x → 0) (ax - (e⁴ˣ - 1))/ax(e⁴ˣ - 1) exists and is equal to b, then the value of… The value of lim _(x → 0) ( x/(⁸√(1 - sin x) - ⁸√(1 + sin x)) ) is equal to : lim _(x → 0) ( (3x² + 2)/(7x² + 2) )^(1/x²) is equal to If lim _(x → 0) (cos (2x) + a cos (4x) - b)/x⁴is finite, then (a + b) is equal to: Let f(x) be a polynomial function such that f(x) + f'(x) + f''(x) = x⁵ + 64. Then, the… More Mathematics chapters Frequently asked questions How many Limits questions are there for JEE Main? PrepWiser has 110 Limits questions for JEE Main, grouped into 5 topics such as Concept of Limit, Standard Limits and L'Hopital's Rule.
Are these Limits questions from previous-year JEE Main papers? Yes — most are genuine previous-year questions from past JEE Main papers, and the rest are practice questions written to match the JEE Main pattern and difficulty.
Where can I see the answers and solutions to these Limits questions? Open any Limits question in PrepWiser (it is free) to attempt it; most come with a step-by-step solution.