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

Question

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 1

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 $\sin x=-1$.

Answer 2

Amplify both sides with $x+\sqrt{1+x^2}$ , and use the fact that $(a-b)(a+b)=a^2-b^2$.