Let [t] denote the greatest integer ≤ t and {t} denote the fractional part of t. The integral value of α for which the left-hand limit of the function \(f(x) = \left\lfloor 1 + x \right\rfloor + \frac{\alpha^{2\left\lfloor x \right\rfloor + \left\{ x \right\}} + \left\lfloor x \right\rfloor - 1}{2\left\lfloor x \right\rfloor + \left\{ x \right\}} \) at x=0 is equal to \(\alpha -\frac{4}{3}\), is
Correct Answer: 3
Solution and Explanation
\(f(x) = \left\lfloor 1 + x \right\rfloor + \frac{\alpha^{2\left\lfloor x \right\rfloor + \left\{ x \right\}} + \left\lfloor x \right\rfloor - 1}{2\left\lfloor x \right\rfloor + \left\{ x \right\}} \)
\(\lim_{{x \to 0^-}} f(x) = \alpha - \frac{4}{3}\)
\(⇒\) \(\lim_{{x \to 0^-}} \left[ 1 + \left\lfloor x \right\rfloor + \frac{\alpha^{x + \left\lfloor x \right\rfloor} + \left\lfloor x \right\rfloor - 1}{x + \left\lfloor x \right\rfloor} \right] = \alpha - \frac{4}{3}\)
\(⇒\) \(\lim_{{h \to 0^-}} \left[ 1 - 1 + \frac{\alpha^{-h - 1} - 1 - 1}{-h - 1} \right] = \alpha - \frac{4}{3}\)
\(∴\) \(\frac{\alpha^{-1} - 2}{-1} = \alpha - \frac{4}{3}\)
\(⇒\) \(3α^2 – 10α + 3 = 0\)
\(∴\) \(α = 3 \ or\ \frac{1}{3}\)
\(∵\) α in integer, hence \(α = 3\)
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main limits and derivatives Questions
- Let \(\{x\}\) denote the fractional part of \(x\), and \(f(x) = \frac{\cos^{-1}(1 - \{x\}^2) \sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3}, \quad x \neq 0\).If \(L\) and \(R\) respectively denote the left-hand limit and the right-hand limit of \(f(x)\) at \(x = 0\), then \(\frac{32}{\pi^2} \left(L^2 + R^2\right)\) is equal to \(\_\_\_\_\_\_\_\_\).
- Let \( P = \{ z \in \mathbb{C} : |z + 2 - 3i| \leq 1 \} \) and \( Q = \{ z \in \mathbb{C} : z(1 + i) + \overline{z}(1 - i) \leq -8 \} \).
Let \( z \) in \( P \cap Q \) have \( |z - 3 + 2i| \) be maximum and minimum at \( z_1 \) and \( z_2 \), respectively.
If \( |z_1|^2 + 2|z_2|^2 = \alpha + \beta \sqrt{2} \), where \( \alpha \) and \( \beta \) are integers, then \( \alpha + \beta \) equals ____. \(\lim_{x \to 0} \frac{e - (1 + 2x)^{\frac{1}{2x}}}{x} \quad \text{is equal to:}\)
- The integral \(\int_{0}^{\pi/4} \frac{136 \sin x}{3 \sin x + 5 \cos x} \, dx\) is equal to
- The value of \(\int_{-\pi}^{\pi} \frac{2y(1 + \sin y)}{1 + \cos^2 y} \, dy\)
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series 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