1

What is the value of this limit :

$$\lim_{n \rightarrow \infty} \frac{n ^ {(1 + \sin{(n)})}} {\sqrt{n}}$$

The original question was to find out the asymptotic relation between $n^{(1 + \sin{(n)})}$ and $\sqrt{n}$.

Swapna Sarit
  • 19
  • 1

1 Answers1

3

For infinitely many $n$, $\sin n < -3/4$, so along this subsequence, $$\frac{n^{1+\sin n}}{\sqrt{n}} ≤ n^{-1/4} → 0.$$

However, for infinitely many other $n$, $\sin n > 0$, so here we see $$ \frac{n^{1+\sin n}}{\sqrt n} ≥ n^{1/2} → ∞. $$

Calvin Khor
  • 34,903
  • Can you please explain?how this statement coming? For infinitely many $n, sinn<−3/4$ so along this subsequence, –  Aug 08 '17 at 14:11
  • @Maneesh.Maths are you asking why ${\sin(n) : n≥0}$ is dense in $[-1,1]$? I suggest reading the answers here https://math.stackexchange.com/questions/4764/sine-function-dense-in-1-1 – Calvin Khor Aug 08 '17 at 14:15
  • Now I understood. I forgot this result.thank you. –  Aug 08 '17 at 14:23
  • You do not need the density result. All you need is for $e^{in}$ to land infinitely many times in the arc centered at $(-1,0)$ having length $>\pi/3.$ – zhw. Aug 08 '17 at 16:59
  • @zhw. You are of course right, that was just the first related question I found that had good answers. – Calvin Khor Aug 08 '17 at 19:01