Evaluate $\lim\limits_{x\to+\infty}\frac{\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n}{x^n},n\in \mathbb{N}$ If we use the following
$$a^n-b^n=\left(a-b\right)\left(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1}\right)=u\times t$$
$$u=x+\sqrt{x^2-1}-x+\sqrt{x^2-1}=2\sqrt{x^2-1}=2x\sqrt{1-\frac{1}{x}}$$
Now, the limit is
$$2\lim\limits_{x\to+\infty}\frac{x\sqrt{1-\frac{1}{x}}\times t}{x^n}$$
What to do next, if this is a good approach? 

Correction to your answer: $$2\sqrt{x^2-1}=2x\sqrt{1-\frac{1}{x^2}}$$
Your approach is going to have trouble because $t$ is a fairly complicated mess.
Note:
$$\frac{x+\sqrt{x^2-1}}{x} = 1+\sqrt{1-\frac{1}{x^2}}\to 2\text{ as } x\to +\infty$$
and
$$\frac{x-\sqrt{x^2-1}}{x}= 1-\sqrt{1-\frac{1}{x^2}}\to 0\text{ as }x\to+\infty$$
