Find the Limit of X and Y Exponents as n Goes to Infinity

[jinbaw]

Homework Statement

$$X \geq Y > 0, find \lim_{n \to \infty} \left(\frac{2X^n + 7Y^n}{2}\right)^{1/n}$$

Homework Equations

The Attempt at a Solution

I'm not really sure how to do it, but i guess I need to use the fact that $$\frac{Y}{X} \leq 1$$, and so $$\lim_{n \to \infty} \left(\frac{Y}{X}\right)^n = 0.$$
Thanks for your help.

 

[Cyosis]

Multiply the argument by $$\frac{X^n}{X^n}$$ then take the $$X^n$$ out of the brackets.

 

[jinbaw]

$$\lim_{n \to \infty} X \left( 1 + \left( \frac{7Y}{2X} \right)^n \right)^{1/n}$$
But I'm still stuck on how to procceed, if you could help.

 

[Cyosis]

There is one mistake, 7/2 should not be within the brackets. Now take X in front of the limit and use $$\frac{Y}{X} \leq 1 \Rightarrow \lim_{n \to \infty} \left(\frac{Y}{X}\right)^n = 0$$.

 

[jinbaw]

Oh right.. So its limit is X.
Thanks :-D

 

[Cyosis]

That's right!