lim _ n ( n^2 + n ) , where n I , is - Vidyadip

Question

$\mathop {lim}\limits_{n \to \infty } \cos \left( {\pi \sqrt {{n^2} + n} } \right)$ , where $n \in I$ , is

✓

Answer

c
$\mathop {\lim }\limits_{n \to \infty } \pm \cos \left( {n\pi - \pi \sqrt {{n^2} + n} } \right)$

$\mathop {\lim }\limits_{n \to \infty } \pm \cos \pi \left[ {\frac{{{n^2} - \left( {{n^2} + n} \right)}}{{n + \sqrt {{n^2} + n} }}} \right]$

$\mathop {\lim }\limits_{n \to \infty } \pm \cos \left( {\pi \times \frac{{ - n}}{{n + \sqrt {{n^2} + n} }}} \right)$

$ \pm \cos \left( {\pi \times \frac{{ - 1}}{2}} \right) = 0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $S=\{1,2,3,4,5,6\}$. Then the number of oneone functions $f: S \rightarrow P(S)$, where $P(S)$ denote the power set of $S$, such that $f(n) \subset f(m)$ where $n < m$ is $..................$

Let $a_{1}, a_{2}, \ldots \ldots, a_{21}$ be an $A.P.$ such that $\sum_{n=1}^{20} \frac{1}{a_{n} a_{n+1}}=\frac{4}{9}$. If the sum of this AP is $189,$ then $a_{6} \mathrm{a}_{16}$ is equal to :

Let $S_k, k=1,2, \ldots ., 100$, denote the sum of the infinite geometric series whose first term is $\frac{k-1}{k!}$ and the common ratio is $\frac{1}{k}$. Then the value of $\frac{100^2}{100!}+\sum_{k=1}^{100}\left|\left(\mathrm{k}^2-3 \mathrm{k}+1\right) \mathrm{S}_{\mathrm{k}}\right|$ is

Let the foci of a hyperbola $\mathrm{H}$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $\mathrm{H}$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $\mathrm{H}$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha^2+2 \beta^2$ is equal to :

The value of $'a'$ $(a>0)$ for which the area bounded by the curves $y = \frac{x}{6} + \frac{1}{{{x^2}}},\,y = 0,\,x = a$ and $x = 2a$ has the least value, is

The number of ordered pairs $(x, y)$ of integers satisfying $x^3+y^3=65$ is

Let $\mathrm{P}(\alpha, \beta)$ be a point on the parabola $\mathrm{y}^2=4 \mathrm{x}$. If $\mathrm{P}$ also lies on the chord of the parabola $x^2=8 y$ whose mid point is $\left(1, \frac{5}{4}\right)$. Then $(\alpha-28)(\beta-8)$ is equal to...................

A $G.P.$ consists of an even number of terms. If the sum of all the terms is $5$ times the sum of the terms occupying odd places, then the common ratio will be equal to

Order of the differential equation of the family of all concentric circles centered at $(h, k)$ is

The function ${x^5} - 5{x^4} + 5{x^3} - 10$ has a maximum, when $x =$