Let x_n= (2^n+3^n )^ 1 / 2 n for all natural numbers n. Then, - Vidyadip

MCQ

Let $x_n=\left(2^n+3^n\right)^{1 / 2 n}$ for all natural numbers $n$. Then,

A
$\lim _{n \rightarrow \infty} x_n=\infty$
✓
$\lim _{n \rightarrow \infty} x_n=\sqrt{3}$
C
$\lim _{n \rightarrow \infty} x_n=\sqrt{3}+\sqrt{2}$
D
$\lim _{n \rightarrow \infty} x_n=\sqrt{5}$

✓

Answer

Correct option: B.

$\lim _{n \rightarrow \infty} x_n=\sqrt{3}$

b
(b)

We have,

$x_n =\left(2^n+3^n\right)^{\frac{1}{2 n}}$

$\Rightarrow \quad \lim _{n \rightarrow \infty} x_n =\lim _{n \rightarrow \infty}\left(2^n+3^n\right)^{\frac{1}{2 n}}$

$=\lim _{n \rightarrow \infty} 3^{n / 2 n}\left[\left(\frac{2}{3}\right)^n+1\right]^{\frac{1}{2 n}}$

$=3^{3 / 2} \times 1=\sqrt{3}$

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

More "MCQ" from 12. limits→12. limits — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

For every positive integer n, $\text{x},\frac{\text{n}^7}{7}+\frac{\text{n}^6}{5}+\frac{2\text{n}^3}{3}-\frac{\text{x}}{105}$ is:

The base $BC$ of a triangle $ABC$ is bisected at the point $(p, q)$ and the equations to the sides $AB$ and $AC$ are respectively $px+qy= 1$ and $qx + py = 1.$ Then the equation to the median through $A$ is

Let $f(x)={{x}^{2}}-x+k-2,k\in R$ then the complete set of values of $k$ for which $y=\left| f\left( \left| x \right| \right) \right|$ is non-derivable at $5$ distinict points is

Given a sequence of $4$ numbers, first three of which are in $G.P.$ and the last three are in $A.P$. with common difference six. If first and last terms in this sequence are equal, then the last term is

${\sin ^2}\frac{\pi }{8} + {\sin ^2}\frac{{3\pi }}{8} + {\sin ^2}\frac{{5\pi }}{8} + {\sin ^2}\frac{{7\pi }}{8} = $

If $S=\{z \in C:|z-i|=|z+i|=|z-1|\}$, then, $n(S)$ is:

The co-ordinates of the vertices $A$ and $B$ of an isosceles triangle $ABC (AC = BC)$ are $(-2,3)$ and $(2,0)$ respectively. $A$ line parallel to $AB$ and having a $y$ -intercept equal to $\frac{43}{12}$ passes through $C$, then the co-ordinates of $C$ are :-

The value of ${(1 + i)^6} + {(1 - i)^6}$ is

The number of terms in the series $101 + 99 + 97 + ..... + 47$ is

Angle between the lines $2x - y - 15 = 0$ and $3x + y + 4 = 0$ is .....$^o$