JEE Mains · Maths · STD 11 - 12. limits
\(\mathop {\lim }\limits_{x \to 0} \frac{{x + 2\,\sin \,x}}{{\sqrt {{x^2} + 2\sin \,x + 1} - \sqrt {{{\sin }^2}\,x - x + 1} }}\) is
- A \(2\)
- B \(6\)
- C \(3\)
- D \(1\)
Answer & Solution
Correct Answer
(A) \(2\)
Step-by-step Solution
Detailed explanation
\(\mathop {\lim }\limits_{x \to 0} \frac{{x + 2\sin x}}{{\sqrt {{x^2} + 2\sin x + 1} - \sqrt {{{\sin }^2}x - x + 1} }}\) \( = \mathop {\lim }\limits_{x \to 0} \frac{{x + 2\sin x}}{{{x^2} + 2\sin x + 1 - {{\sin }^2}x - x + 1}}\)…
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 triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon isJEE Mains 2024 Medium
- Ifa variable plane, at a distance of \(3\, units\) from the origin, intersects the coordinate axes at \(A, B\) and \(C\), then the locus of the centroid of \(\Delta ABC\) isJEE Mains 2017 Hard
- Let \(a, b \in R\). If the mirror image of the point \(P( a ,6,9)\) with respect to the line \(\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}\) is \((20, b,-a-9),\) then \(|a+b|\) is equal to :JEE Mains 2021 Hard
- Let \(y=y(x)\) be the solution of the differential equation \(x\frac{dy}{dx}-y=x^{2}\cot x, x\in(0,\pi)\). If \(y(\frac{\pi}{2})=\frac{\pi}{2}\), then \(6y(\frac{\pi}{6})-8y(\frac{\pi}{4})\) is equal to :JEE Mains 2026 Easy
- The equation of a plane through the line of intersection of the planes \(x + 2y = 3,y-2z + 1= 0\), and perpendicular to the first plane isJEE Mains 2013 Hard
- Let \((a, b)\) be the point of intersection of the curve \(x^2=2 y\) and the straight line \(y-2 x-6=0\) in the second quadrant. Then the integral \(I=\int_a^b \frac{9 x^2}{1+5^x} d x\) is equal to :JEE Mains 2025 Medium
More PYQs from JEE Mains
- Let the ellipse \(3 x^2+\mathrm{py}^2=4\) pass through the centre \(C\) of the circle \(x^2+y^2-2 x-4 y-11=0\) of radius \(r\). Let \(f_1, f_2\) be the focal distances of the point C on the ellipse. Then \(6 f_1 f_2-r\) is equal toJEE Mains 2025 Medium
- Let \(P\) be the plane passing through the intersection of the planes
\(\overrightarrow{ r } \cdot(\hat{ i }+3 \hat{ j }-\hat{ k })=5\) and \(\overrightarrow{ r } \cdot(2 \hat{ i }-\hat{ j }+\hat{ k })=3\), and the point \((2,1,-2)\). Let the position vectors of the points \(X\) and \(Y\) be \(\hat{ i }-2 \hat{ j }+4 \hat{ k }\) and \(5 \hat{ i }-\hat{ j }+2 \hat{ k }\) respectively. Then the points
JEE Mains 2022 Medium - Let the circles \(C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2\) and \(C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2\) touch each other externally at the point \((6,6)\). If the point \((6,6)\) divides the line segment joining the centres of the circles \(C_1\) and \(C_2\) internally in the ratio \(2: 1\), then \((\alpha+\beta)+4\left(r_1^2+r_2^2\right)\) equalsJEE Mains 2024 Hard
- Let \(B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]\) and \(A\) be a \(2 \times 2\) matrix such that \(\mathrm{AB}^{-1}=\mathrm{A}^{-1}\). If \(\mathrm{BCB}^{-1}=\mathrm{A}\) and \(\mathrm{C}^4+\alpha \mathrm{C}^2+\beta \mathrm{I}=\mathrm{O}\), then \(2 \beta-\alpha\) is equal to :JEE Mains 2024 Hard
- If \(I(x)=\int e^{\sin ^2 x}(\cos x \sin 2 x-\sin x) d x \quad\) and \(I(0)=1\), then \(I\left(\frac{\pi}{3}\right)\) is equal toJEE Mains 2023 Hard
- If \({\left( {10} \right)^9} + 2{\left( {11} \right)^1}{\left( {10} \right)^8} + 3{\left( {11} \right)^2}{\left( {10} \right)^7} + ..\;.\;.\;.\; + 10\left( {{{11}^9}} \right) = \;k{\left( {10} \right)^9}\) ,then \(k \) is equal toJEE Mains 2014 Hard