lim-n-cosn-sinn-3-n-4-n-

Question Number 206702 by depressiveshrek last updated on 22/Apr/24

\lim_{n \to \infty} \sqrt{\cos n + \sin n - 3^{n} + 4^{n}}

Answered by Frix last updated on 22/Apr/24

- \sqrt{2} ⩽ \cos n + \sin n ⩽ \sqrt{2}

\forall a \in R : \lim_{n \to \infty} \sqrt{4^{n} - 3^{n} + a} = \lim_{n \to \infty} \sqrt{4^{n} - 3^{n}} = \infty

Answered by mathzup last updated on 23/Apr/24

u_{n} = \sqrt{c o s n + s i n n + 4^{n} - 3^{n}}

\Rightarrow u_{n} = \sqrt{\sqrt{2} c o s (n + \frac{π}{4}) + 4^{n} - 3^{n}}

= 2^{n} \sqrt{\frac{\sqrt{2} c o s (n + \frac{π}{4})}{4^{n}} + 1 - {(\frac{3}{4})}^{n}}

\Rightarrow u_{n} \sim 2^{n} \sqrt{1 - {(\frac{3}{4})}^{n}} \sim 2^{n} \Rightarrow

{l i m}_{n \to + \infty} u_{n} = + \infty

Facebook Tweet Pin