The value of the limit
$\displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3 x+\sin x)}{\left(2 \sin 2 x \sin \frac{3 x}{2}+\cos \frac{5 x}{2}\right)-\left(\sqrt{2}+\sqrt{2} \cos 2 x+\cos \frac{3 x}{2}\right)}$
is _______
$\displaystyle\lim _{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3 x+\sin x)}{\left(2 \sin 2 x \sin \frac{3 x}{2}+\cos \frac{5 x}{2}\right)-\left(\sqrt{2}+\sqrt{2} \cos 2 x+\cos \frac{3 x}{2}\right)}$
is _______
Correct Answer: 8
Solution and Explanation
Answer is 8.
Top Questions on Limits
Consider two statements:
Statement 1: $ \lim_{x \to 0} \frac{\tan^{-1} x + \ln \left( \frac{1+x}{1-x} \right) - 2x}{x^5} = \frac{2}{5} $
Statement 2: $ \lim_{x \to 1} x \left( \frac{2}{1-x} \right) = e^2 \; \text{and can be solved by the method} \lim_{x \to 1} \frac{f(x)}{g(x) - 1} $- Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]
- Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]
- Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]
- Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]
Questions Asked in JEE Advanced exam
A positive, singly ionized atom of mass number $ A_M $ is accelerated from rest by the voltage $ 192 \, \text{V} $. Thereafter, it enters a rectangular region of width $ w $ with magnetic field $ \vec{B}_0 = 0.1\hat{k} \, \text{T} $. The ion finally hits a detector at the distance $ x $ below its starting trajectory. Which of the following option(s) is(are) correct?
$ \text{(Given: Mass of neutron/proton = } \frac{5}{3} \times 10^{-27} \, \text{kg, charge of the electron = } 1.6 \times 10^{-19} \, \text{C).} $
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
- Let \(\overrightarrow{OP}=\frac{\alpha-1}{\alpha}\hat{i}+\hat{j}+\hat{k},\overrightarrow{OQ}=\hat{i}+\frac{\beta-1}{\beta}\hat{j}+\hat{k}\) and \(\overrightarrow{OR}=\hat{i}+\hat{j}+\frac{1}{2}\hat{k}\) be three vector where α, β ∈ R - {0} and 0 denotes the origin. If \((\overrightarrow{OP}\times\overrightarrow{OQ}).\overrightarrow{OR}=0\) and the point (α, β, 2) lies on the plane 3x + 3y - z + l = 0, then the value of l is _______.
- JEE Advanced - 2024
- Vector Algebra
- Let \(S=\left\{(x,y)\in\R\times\R:x\ge0,y\ge0,y^2\le12-2x\ \text{and}\ 3y+\sqrt8\ x\le5\sqrt8\right\}\). If the area of the region S is \(\alpha\sqrt2\), then α is equal to
- JEE Advanced - 2024
- Coordinate Geometry
- Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.
- JEE Advanced - 2024
- Matrices
- A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and is in a uniform vertical magnetic field \( B_0 \). When a current \( I \) is passed through the loop, the loop turns about the line PQ by an angle \( \theta \). The angle \( \theta \) is given by:
- JEE Advanced - 2024
- Electromagnetic Field (EMF)
Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).