I'm having a really tough time trying to evaluate the limit for this expression
$\lim_{x\to\infty}\frac{\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}$
The only hint I was given is that $\lim_{x\to\infty}\frac{\sin(x)}{x} = 0$.
By using trigonometric identities I can see that this leads to $\lim_{x\to\infty}\frac{\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{\sin^2(\frac{1}{x})}$. I proceeded then to do multiply everything by $\frac{1}{x}$ which leaves me with $\lim_{x\to\infty}\frac{x^{-1}\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{1}$ (according to the hint I was given).
However, now I don't know how to simplify the expression in the numerator. Any tips would be really appreciated.
Thanks