For positive integer $n$, define $f(n)=n+\frac{16+5 n-3 n^2}{4 n+3 n^2}+\frac{32+n-3 n^2}{8 n+3 n^2}+\frac{48-3 n-3 n^2}{12 n+3 n^2}+\ldots+\frac{25 n-7 n^2}{7 n^2}$.
Then, the value of $\displaystyle\lim _{n \rightarrow \infty} f(n)$ is equal to
Then, the value of $\displaystyle\lim _{n \rightarrow \infty} f(n)$ is equal to
- $3+\frac{4}{3} \log _e 7$
- $4-\frac{3}{4} \log _e\left(\frac{7}{3}\right)$
- $4-\frac{4}{3} \log _e\left(\frac{7}{3}\right)$
- $3+\frac{3}{4} \log _{ e } 7$
The Correct Option is B
Approach Solution - 1
The correct option is (B): \(4-\frac{3}{4} \log _e\left(\frac{7}{3}\right)\)
Approach Solution -2
We are given the function:
\(f(n) = n + \frac{16 + 5n - 3n^2}{4n + 3n^2} + \cdots + \frac{25n - 7n^2}{7n^2}\)
\(= \left(\frac{16 + 5n - 3n^2}{4n + 3n^2} + 1\right) + \left(\frac{32 + n - 3n^2}{8n + 3n^2} + 1\right) + \cdots + \left(\frac{25n - 7n^2}{7n^2} + 1\right)\)
We need to find the limit:
\(\lim_{n \to \infty}f(n)\)
Now let us simplify
\(f(n) = \frac{9n + 16}{4n + 3n^2} + \frac{9n + 32}{8n + 3n^2} + \cdots + \frac{25n}{7n^2}\)
\(= \sum_{r=1}^{n} \frac{9n + 16r}{4rn + 3n^2} = \frac{1}{n} \sum_{r=1}^{n} \frac{9 + 16\left(\frac{r}{n}\right)}{4 \left(\frac{r}{n}\right) + 3}\)
\(\lim_{n \to \infty} f(n) = \int_{0}^{1} \frac{9 + 16x}{4x + 3} \, dx\)
\(= \int_{0}^{1} \left(\frac{16x + 12}{4x + 3} - 3\right) \, dx\)
\(= \left[4x - \frac{3}{4} \ln |4x + 3|\right]_{0}^{1}\)
\(=4-\frac{3}{4} \log _e\left(\frac{7}{3}\right)\)
So, the correct option is (B): \(4-\frac{3}{4} \log _e\left(\frac{7}{3}\right)\)
Top Questions on Limits
- If \( f(x) = \begin{cases} \frac{(e^x - 1) \log(1 + x)}{x^2} & \text{if } x>0 \\ 1 & \text{if } x = 0 \\ \frac{\cos 4x - \cos bx}{\tan^2 x} & \text{if } x<0 \end{cases} \) is continuous at \( x = 0 \), then \(\sqrt{b^2 - a^2} =\)
- \(\lim_{x \to 0} \frac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2} =\)
- Let \([x]\) represent the greatest integer function. If \(\lim_{x \to 0^+} \frac{\cos[x] - \cos(kx - [x])}{x^2} = 5\), then \(k =\)
- Evaluate the limit: \[ \lim_{x \to 0} \frac{(\csc x - \cot x)(e^x - e^{-x})}{\sqrt{3} - \sqrt{2 + \cos x}} \]
- If a real valued function \( f(x) = \begin{cases} (1 + \sin x)^{\csc x} & , -\frac{\pi}{2} < x < 0 \\ a & , x = 0 \\ \frac{e^{2/x} + e^{3/x}}{ae^{2/x} + be^{3/x}} & , 0 < x < \frac{\pi}{2} \end{cases} \) is continuous at \(x = 0\), then \(ab = \)
Questions Asked in JEE Advanced exam
- Let $ \mathbb{R} $ denote the set of all real numbers. Let $ a_i, b_i \in \mathbb{R} $ for $ i \in \{1, 2, 3\} $.
Define the functions $ f: \mathbb{R} \to \mathbb{R},\ g: \mathbb{R} \to \mathbb{R},\ h: \mathbb{R} \to \mathbb{R} $ by:
$$ f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4,\quad g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4, $$ $$ h(x) = f(x+1) - g(x+2) $$ If $ f(x) \ne g(x) $ for every $ x \in \mathbb{R} $, then the coefficient of $ x^3 $ in $ h(x) $ is:- JEE Advanced - 2025
- Polynomials
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
- JEE Advanced - 2025
- Differential equations
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
- JEE Advanced - 2025
- Coordinate Geometry
- A projectile is thrown at an angle of \(60^\circ\) with the horizontal. Initial speed is \(270\, \text{m/s}\). A linear drag force \(F = -CV\) acts on the body. Find the horizontal displacement till \(t = 2\) seconds. Given \(C = 0.1\, \text{s}^{-1}\).
- JEE Advanced - 2025
- Projectile motion
- A single slit diffraction experiment is performed to determine the slit width using the equation, $ \dfrac{b d}{D} = m \lambda $, where $ b $ is the slit width, $ D $ the shortest distance between the slit and the screen, $ d $ the distance between the $ m^{\text{th}} $ diffraction maximum and the central maximum, and $ \lambda $ is the wavelength. $ D $ and $ d $ are measured with scales of least count of 1 cm and 1 mm, respectively. The values of $ \lambda $ and $ m $ are known precisely to be $ 600 \, \text{nm} $ and 3, respectively. The absolute error (in $ \mu\text{m} $) in the value of $ b $ estimated using the diffraction maximum that occurs for $ m = 3 $ with $ d = 5 \, \text{mm} $ and $ D = 1 \, \text{m} $ is ___.
- JEE Advanced - 2025
- Error Propagation
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).