Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20244 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Limits

JEE Main 2024 · 4 April · Evening Shift · Mathematics · Limits

Let f(x)=∫_0^x(t+ (1-e^ )) d t, x ∈ ℝ . Then, _ x arrow 0...

Let f(x)=∫0x(t+sin⁡(1−e′))dt,x∈Rf(x)=\int_0^x\left(t+\sin \left(1-e^{\prime}\right)\right) d t, x \in \mathbb{R}. Then, lim⁡x→0f(x)x3\lim _{x \rightarrow 0} \frac{f(x)}{x^3} is equal to

Options

  1. A−16-\frac{1}{6}
  2. B23\frac{2}{3}
  3. C−23-\frac{2}{3}
  4. D16\frac{1}{6}

Correct answer

A. −16-\frac{1}{6}

Step-by-step solution

lim⁡x→0f(x)x3\lim _{x \rightarrow 0} \frac{f(x)}{x^3}
Using L Hopital Rule. lim⁡x→0f′(x)3x2=lim⁡x→0x+sin⁡(1−ex)3x2\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{3 x^2}=\lim _{x \rightarrow 0} \frac{x+\sin \left(1-e^x\right)}{3 x^2} (Again L Hopital) Using L.H. Rule =lim⁡x→0−[sin⁡(1−ex)(−ex)⋅ex+cos⁡(1−ex)⋅ex]6=−16\begin{aligned} & =\lim _{x \rightarrow 0} \frac{-\left[\sin \left(1-e^x\right)\left(-e^x\right) \cdot e^x+\cos \left(1-e^x\right) \cdot e^x\right]}{6} \\ & =-\frac{1}{6} \end{aligned}
Practice Limits on Quantrex Academy →

More from Limits

Let _ x → 2 (( (x-2))(rx^2 + (p-2)x - 2p))/((x-2)^2) = 5 for some r, p ∈ ℝ . If the set… JEE Main 2026The value of _ x → 0 ((x^2 ^2 x)/(x^2 - ^2 x)) is: JEE Main 2026Let f(x) = _ y → 0 ((1 - (xy)) (xy))/(y^3) . Then the number of solutions of the equation… JEE Main 2026The product of all possible values of α , for which _ x → 0 ((1 - (α x) ((α+1)x)… JEE Main 2026If _ x → 2 (x^3 - 5x^2 + ax + b) ( x-1 - 1) _e(x-1) = m , then a + b + m is equal to : JEE Main 2026The value of _ x arrow 0 _ e ( (e x) · (e^ 2 x) · … · (e^ 10 x)) e^ 2 -e^ 2 x is equal to JEE Main 2026If _ x arrow 0 e^ (a-1) x +2 ~b x+(c-2) e^ -x x x- _ e (1+x) =2 , then a^ 2 +b^ 2 +c^ 2… JEE Main 2026Let [·] denote the greatest integer function and f(x)= _ n arrow ∞ 1 n^ 3 ∑_ k=1 ^ n [ k^… JEE Main 2026Let f: R arrow(0, ∞) be a twice differentiable function such that f(3)=18, f^ (3)=0 and… JEE Main 2026Let α and β be the real numbers such that _ x arrow 0 (1)/(x^3)((α)/(2) ∫_0^x (1)/(1-t^2)… JEE Advanced 2025Let x_0 be the real number such that e^ x_0 +x_0=0 . For a given real number α , define… JEE Advanced 2025Given below are two statements : Statement I : _ x arrow 0 ( ^ -1 x+ _e (1+x)/(1-x) -2 x… JEE Main 2025If the function f(x)=( ( x)- ( x))/( x- x) is continuous at x=0 , then f(0) is equal to… JEE Main 2025_ x arrow 0^ + (5(x)^ (1)/(3) ) _e(1+3 x^2) ( ^ -1 3 x )^2(e^ 5(x)^ (4)/(3) -1) is equal… JEE Main 2025If _ x arrow 1^ + ((x-1)(6+λ (x-1))+μ (1-x))/((x-1)^3)=-1 , where λ, μ ∈ ℝ , then λ+μ is… JEE Main 2025If Lim_ x arrow 0 (( x)/(x))^ (1)/(x^2) =p , then 96 _e p is equal to ______ JEE Main 2025 Full Limits list Limits JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs