1. ## Solving this limit

Find n such that
$\displaystyle \lim_{x\rightarrow\infty} \frac {((nx+1)^x)}{(nx+1)^x}= 9$

i know the question ends up as e^(-2/n) but can someone show me a different way of solving it other than the fact that
$\displaystyle e^a = \lim_{x\rightarrow\infty} (1+ (a/x))^x$ ?

i attempted to solve it with ln, moving the x's down but it would be great if i could get some assistance

Edit: sorry i couldn't get the language for this math down, any help on the actual problem? XD sorry

2. If your question is find n such that $\displaystyle \lim_{x \to \infty}(nx+1)^x=9$ no such n (with n a real number) exists

Now if we let n be a polynomial, well we already know that $\displaystyle \lim_{x \to \infty}{(\frac{y}{x}+1)^x}=e^y$ so what y will make $\displaystyle e^y=9$ this is a simple equation and you can probably see that y must be 2ln3 or ln9

So we need nx to be $\displaystyle \frac{ln9}{x}$ so n must be $\displaystyle \frac{ln9}{x^2}$

3. Soo Sorry, i wrote the question out wrong, could i get some assistance on the actual problem now? Again, i didn't know the language for making the site draw sigma, sorry.