If $a =\displaystyle\lim _{ n \rightarrow \infty} \sum_{ k =1}^{ n } \frac{2 n }{ n ^2+ k ^2}$ and $f(x)=\sqrt{\frac{1-\cos x}{1+\cos x}}, x \in(0,1)$, then :
- $2 \sqrt{2} f \left(\frac{ a }{2}\right)= f ^{\prime}\left(\frac{ a }{2}\right)$
- $f\left(\frac{a}{2}\right) f^{\prime}\left(\frac{a}{2}\right)=\sqrt{2}$
- $\sqrt{2} f \left(\frac{ a }{2}\right)= f ^{\prime}\left(\frac{ a }{2}\right)$
- $f \left(\frac{ a }{2}\right)=\sqrt{2} f ^{\prime}\left(\frac{ a }{2}\right)$
The Correct Option is C
Solution and Explanation
The correct option is (C): $\sqrt{2} f \left(\frac{ a }{2}\right)= f ^{\prime}\left(\frac{ a }{2}\right)$
Top Questions on limits and derivatives
Let $\left\lfloor t \right\rfloor$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbb{N}$ for which
\[ \lim_{x \to 0^+} \left( x \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \dots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \dots + \left\lfloor \frac{9^2}{x^2} \right\rfloor \right) \geq 1 \]
is equal to __________.
- JEE Main - 2025
- Mathematics
- limits and derivatives
- Find the derivative: \[ \frac{d}{dx} \left[ \left| \frac{x}{4} \right| \right] \]
- Bihar Board XII - 2025
- Mathematics
- limits and derivatives
- Find the derivative of \[ \frac{d}{dx} \left( \lim_{n \to 1} \frac{x^n - 1}{n+1} \right). \]
- Bihar Board XII - 2025
- Mathematics
- limits and derivatives
- Find the derivative of \( \log(x^9) \).
- Bihar Board XII - 2025
- Mathematics
- limits and derivatives
- \(\frac{d}{dx} \left[ \log x^2 + \log a^2 \right] \)
- Bihar Board XII - 2025
- Mathematics
- limits and derivatives
Questions Asked in JEE Main exam
Considering Bohr’s atomic model for hydrogen atom :
(A) the energy of H atom in ground state is same as energy of He+ ion in its first excited state.
(B) the energy of H atom in ground state is same as that for Li++ ion in its second excited state.
(C) the energy of H atom in its ground state is same as that of He+ ion for its ground state.
(D) the energy of He+ ion in its first excited state is same as that for Li++ ion in its ground state.- JEE Main - 2025
- Atomic Models
- In an adiabatic process, which of the following statements is true?
- JEE Main - 2025
- Thermodynamics
- To obtain the given truth table, the following logic gate should be placed at G
- JEE Main - 2025
- Logic gates
- Choose the correct logic circuit for the given truth table having inputs A and B.
- JEE Main - 2025
- Logic gates
A slanted object AB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is:
Concepts Used:
Limits And Derivatives
Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.
Limits Formula:
Derivatives of a Function:
A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.
Properties of Derivatives:
Read More: Limits and Derivatives