JEE Mains · Maths · STD 11 - 12. limits
Let \(f(x)=\int_0^x\left(t+\sin \left(1-e^t\right)\right) d t, x \in \mathbb{R}\). Then \(\lim _{x \rightarrow 0} \frac{f(x)}{x^3}\) is equal to
- A \(\frac{1}{6}\)
- B \(-\frac{1}{6}\)
- C \(-\frac{2}{3}\)
- D \(\frac{2}{3}\)
Answer & Solution
Correct Answer
(B) \(-\frac{1}{6}\)
Step-by-step Solution
Detailed explanation
\(\lim _{x \rightarrow 0} \frac{f(x)}{x^3}\) Using L Hopital Rule. \(\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{3 x^2}=\lim _{x \rightarrow 0} \frac{x+\sin \left(1-e^x\right)}{3 x^2}\) (Again L Hopital) Using \(L.H.\) Rule…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The number of ways of selecting \(15\) teams from \(15\) men and \(15\) women, such that each team consists of a man and a woman, isJEE Mains 2015 Hard
- Let \(y = y(x)\) be the solution of the differential equation \((\tan x)^{1/2}\,dy = (\sec^3 x - (\tan x)^{3/2} y)\,dx\), \(0 < x < \dfrac{\pi}{2}\), \(y\left(\dfrac{\pi}{4}\right) = \dfrac{6\sqrt{2}}{5}\). If \(y\left(\dfrac{\pi}{3}\right) = \dfrac{4}{5}\alpha\), then \(\alpha^4\) equals _______.JEE Mains 2026 Hard
- \(\)f(x)=\left| {\begin{array}{*{20}{c}} {{{\sin }^2}x}&{ - 2 + {{\cos }^2}x}&{\cos 2x} \\ {2 + {{\sin }^2}x}&{{{\cos }^2}x}&{\cos 2x} \\ {{{\sin }^2}x}&{{{\cos }^2}x}&{1 + \cos 2x} \end{array}} \right| ,x \in[0, \pi]\(\) Then the maximum value of \(f(x)\) is equal to \(.....\)JEE Mains 2021 Hard
- If \(k=tan(\frac{\pi}{4}+\frac{1}{2}cos^{-1}(\frac{2}{3}))+tan(\frac{1}{2}sin^{-1}(\frac{2}{3}))\) then the number of solutions of the equation \(sin^{-1}(kx-1)=sin^{-1}x-cos^{-1}x\) is ___ .JEE Mains 2026 Hard
- Let \(A =\{ x \in R :| x +1|<2\}\) and \(B=\{x \in R:|x-1| \geq 2\}\). Then which one of the following statements is NOT true ?JEE Mains 2022 Medium
- Locus of the image of point \( (2,3)\) in the line \(\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,k \in R\) is a:JEE Mains 2015 Hard
More PYQs from JEE Mains
- If \(x = x ( y )\) is the solution of the differential equation \(y \frac{d x}{d y}=2 x+y^{3}(y+1) e^{y}, x(1)=0\); then \(x(e)\) is equal toJEE Mains 2022 Hard
- The absolute difference of the coefficients of \(x^{10}\) and \(x^7\) in the expansion of \(\left(2 x^2+\frac{1}{2 x}\right)^{11}\) is equal toJEE Mains 2023 Hard
- All possible values of \(\theta \in[0,2 \pi]\) for which \(\sin 2 \theta+\tan 2 \theta>0\) lie inJEE Mains 2021 Hard
- Let \(A\) and \(B\) be two \(3 \times 3\) non-zero real matrices such that \(AB\) is a zero matrix. Then.JEE Mains 2022 Medium
- The distance of the point \((-1,2,3)\) from the plane \(\vec{r} .(\hat{i}-2 \hat{j}+3 \hat{k})=10\) parallel to the line of the shortest distance between the lines \(\overrightarrow{ r }=(\hat{ i }-\hat{ j })+\lambda(2 \hat{i}+\hat{ k })\) and \(\overrightarrow{ r }=(2 \hat{ i }-\hat{ j })+\mu(\hat{i}-\hat{j}+\hat{ k })\) is :JEE Mains 2023 Hard
- Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{ dy }{ dx }+\left(\frac{2 x ^{2}+11 x +13}{ x ^{3}+6 x ^{2}+11 x +6}\right)\) \(y=\frac{(x+3)}{x+1}, x>-1\), which passes through the point \((0,1)\). Then \(y (1)\) is equal to.JEE Mains 2022 Hard