\(lim _{ n → ∞}\bigg (\frac{n^2}{(n^2+1)(n+1)}+\frac{n^2}{(n^2+4)(n+2)}+\frac{n^2}{(n2+9)(n+3)}.....+ \frac{n^2}{(n^2+n^2)(n+n)}\bigg)\)is equal to
- \(\frac{π}{8}+\frac{1}{4} \;log_e2\)
- \(\frac{π}{4}+\frac{1}{8}\; log_e2\)
- \(\frac{π}{4}-\frac{1}{8}\; log_e2\)
- \(\frac{π}{8}+\frac{1}{8}\; log_e\sqrt2\)
The Correct Option is A
Solution and Explanation
\(lim _{ n → ∞}\bigg (\frac{n^2}{(n^2+1)(n+1)}+\frac{n^2}{(n^2+4)(n+2)}+\frac{n^2}{(n2+9)(n+3)}.....+ \frac{n^2}{(n^2+n^2)(n+n)}\bigg)\)
= \(lim _{n → ∞} ∑^n_{ r=1} \frac{n^2}{(n^2+r^2)(n+r)}\)
= \(lim_{ n → ∞} ∑^n_{ r=1} \frac{1}{\bigg[1+(\frac{r}{n})^2\bigg]\bigg[1+(\frac{r}{n})\bigg]}\)
= \(∫^1_0 \frac{1}{(1+x^2)(1+x)}dx\)
= \(\frac{1}{2} ∫^1_0\bigg [\frac{ 1}{1+x}-\frac{(x-1)}{(1+x^2)}\bigg]dx\)
= \(\frac{1}{2}\bigg[\;In(1+x)-\frac{1}{2}\;In(1+x^2)+tan^{-1}x\bigg]^1_0\)
= \(\frac{1}{2}\bigg[\frac{π}{4}+\frac{1}{2}\;In2\bigg]\)
\(=\frac{π}{8}+\frac{1}{4}\;In2\)
Hence, the correct option is (A): \(\frac{π}{8}+\frac{1}{4} \;log_e2\)
Top Questions on Limits
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Questions Asked in JEE Main exam
Let one focus of the hyperbola \( H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \) be at \( (\sqrt{10}, 0) \) and the corresponding directrix be \( x = \dfrac{9}{\sqrt{10}} \). If \( e \) and \( l \) respectively are the eccentricity and the length of the latus rectum of \( H \), then \( 9 \left(e^2 + l \right) \) is equal to:
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- Coordinate Geometry
- The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is:
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- permutations and combinations
- Two identical symmetric double convex lenses of focal length \( f \) are cut into two equal parts \( L_1, L_2 \) by the AB plane and \( L_3, L_4 \) by the XY plane as shown in the figure respectively. The ratio of focal lengths of lenses \( L_1 \) and \( L_3 \) is:
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).