\(\lim_{x\rightarrow \frac{\pi}{4}}\) 8\(\sqrt2\)−(cosx+sinx)\(^{\frac{7}{\sqrt2}}\)−\(\sqrt2\)sin2x is equal to
Correct Answer: 14
Solution and Explanation
=7(\(\frac{1}{\sqrt2}\)+\(\frac{1}{\sqrt2}\))\(^\frac{5}{2\sqrt2}\)
=\(\frac{7\times 4\sqrt2}{2\sqrt2}\)
=14
Top Questions on Limits
For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:
- Match List-I with List-II:
\[\begin{array}{|c|c|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline \text{(A)}\ \lim_{x\to 0}(1+2x)^{\frac{1}{x}} & \text{(I)}\ e^{6} \\ \hline \text{(B)}\ \lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x} & \text{(II)}\ e^{2} \\ \hline \text{(C)}\ \lim_{x\to 0}(1+5x)^{\frac{2}{x}} & \text{(III)}\ e \\ \hline \text{(D)}\ \lim_{x\to \infty}\left(1+\frac{3}{x}\right)^{2x} & \text{(IV)}\ e^{5} \\ \hline \end{array}\] Choose the correct answer from the options given below: - The value of $\displaystyle \lim_{x \to \infty}\left(1+\frac{2}{3x}\right)^{x}$ is:
- Evaluate: \[ \lim_{x \to 0} \frac{\sin 3x - 3 \sin x}{x^3} \]
- Given below are two statements: Statement I: $\lim_{x \to 0} \left( \frac{\tan^{-1} x + \log_e \sqrt{\frac{1+x}{1-x}} - 2x}{x^5} \right) = \frac{2}{5}$ Statement II: $\lim_{x \to 1} \left( \frac{2}{x^{1-x}} \right) = \frac{1}{e^2}$ In the light of the above statements, choose the correct answer from the options given below
Questions Asked in JEE Main exam
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
- JEE Main - 2025
- Relations and functions
The maximum speed of a boat in still water is 27 km/h. Now this boat is moving downstream in a river flowing at 9 km/h. A man in the boat throws a ball vertically upwards with speed of 10 m/s. Range of the ball as observed by an observer at rest on the river bank is _________ cm. (Take \( g = 10 \, {m/s}^2 \)).
- JEE Main - 2025
- System of Particles & Rotational Motion
- 'X' is the number of acidic oxides among $ VO_2, V_2O_3, CrO_3, V_2O_5 $ and $ Mn_2O_7 $. The primary valency of cobalt in $ [Co(H_2NCH_2CH_2NH_2)_3]_2 (SO_4)_3 $ is Y. The value of X + Y is :
- JEE Main - 2025
- Inorganic chemistry
- The kinetic energy of translation of the molecules in 50 g of CO\(_2\) gas at 17°C is:
- JEE Main - 2025
- The Kinetic Theory of Gases
Given below are two statements I and II.
Statement I: Dumas method is used for estimation of "Nitrogen" in an organic compound.
Statement II: Dumas method involves the formation of ammonium sulfate by heating the organic compound with concentrated H\(_2\)SO\(_4\). In the light of the above statements, choose the correct answer from the options given below:- JEE Main - 2025
- Nitrogen
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).