$\displaystyle \lim _{n \rightarrow \infty}\left[\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\ldots+\frac{1}{2 n}\right]$ is equal to
$\log _e2$
- $\log _e\left(\frac{2}{3}\right)$
- 0
- $\log _e\left(\frac{3}{2}\right)$
The Correct Option is A
Approach Solution - 1
\(=\lim_{n\to\infty} ∑^n_{r=1}\frac{1}{n}(\frac{1}{1+\frac{r}{n}})\)
\(= ∫_0^1 \frac{1}{1+x}dx=[ℓln(1+x]_0^1 =ℓn2\)
Approach Solution -2
\[ \lim_{n \to \infty} \sum_{r=1}^{n} \frac{1}{r + r} = \lim_{n \to \infty} \sum_{r=1}^{n} \frac{1}{1 + r} \] Using integration approximation: \[ \int_1^n \frac{dx}{1 + x} = [\ln(1 + x)]_1^n = \ln(1+n) - \ln(2) \approx \log 2 \]
Top Questions on Limits
- Find the equation of the tangent line to the curve \( y = x^2 - 3x + 2 \) at the point \( (1, 0) \).
- The value of \[ \lim_{x \to 1} \frac{x^4 - \sqrt{x}}{\sqrt{x} - 1} \] is:
- \(\lim_{x \to 0} \frac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2} =\)
- 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 \(\sqrt{b^2 - a^2} =\)
- Let \([x]\) represent the greatest integer function. If \(\lim_{x \to 0^+} \frac{\cos[x] - \cos(kx - [x])}{x^2} = 5\), then \(k =\)
Questions Asked in JEE Main exam
- The ratio of the power of a light source \( S_1 \) to that of the light source \( S_2 \) is 2. \( S_1 \) is emitting \( 2 \times 10^{15} \) photons per second at 600 nm. If the wavelength of the source \( S_2 \) is 300 nm, then the number of photons per second emitted by \( S_2 \) is \_\_\_\_\_ \( \times 10^{14} \).
- JEE Main - 2025
- Units and measurement
- In a group of 3 girls and 4 boys, there are two boys \( B_1 \) and \( B_2 \). The number of ways in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but \( B_1 \) and \( B_2 \) are not adjacent to each other, is:
- JEE Main - 2025
- Permutations
Let \( T_r \) be the \( r^{\text{th}} \) term of an A.P. If for some \( m \), \( T_m = \dfrac{1}{25} \), \( T_{25} = \dfrac{1}{20} \), and \( \displaystyle\sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle\sum_{r=m}^{2m} T_r \) is equal to:
- JEE Main - 2025
- Arithmetic and Geometric Progressions
- Let $ A(4, -2), B(1, 1) $ and $ C(9, -3) $ be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is __________.
- JEE Main - 2025
- Geometry
- The value of \( (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) \) is:
- JEE Main - 2025
- Trigonometric Identities
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).