There is an error.
when x goes inside square root, it becomes x^2.
So correct is
Now proceed.
[imath]
\lim _{x->+\infty} \frac{(x-\sqrt{x^2-1})^n+(x+\sqrt{x^2-1})^n}{x^n}=\lim _{x->+\infty} \frac{(1-\sqrt{x-\frac{1}{x}})^n+(1+\sqrt{x-\frac{1}{x}})^n}{1}=\infty-\infty
[/imath]
this is as far as i can go
i got undefined form
??