calculus - Find this limit without L'hopital Rule : $lim_{xrightarrow +infty}frac{x(1+sin(x))}{x-sqrt{(1+x^2)}}$

Find this limit without l'Hopital rule : $$\lim_{x\rightarrow +\infty}\frac{x(1+ \sin x)}{x-\sqrt{1+x^2}}$$

I tried much but can't get any progress!

Answer

The limit does not exist. Multiply top and bottom by $x+\sqrt{1+x^2}$. The bottom becomes $-1$. As to the new top, it is very big if $\sin x$ is not close to $-1$. However, there are arbitrarily large $x$ such that $\sin x=-1$.