Evaluate $$\lim_{x\to+\infty} \frac{\sqrt{x}(\sin{x}+\sqrt{x}\cos{x})}{x\sqrt{x}-\sin({x\sqrt{x})}}$$
My attempt: $$\lim_{x\to+\infty} \frac{\sqrt{x}(\sin{x}+\sqrt{x}\cos{x})}{x\sqrt{x}-\sin({x\sqrt{x})}}=\lim_{x\to+\infty} \frac{\sin{x}+\sqrt{x}\cos{x}}{x-\frac{\sin({x\sqrt{x})}}{\sqrt{x}}}$$
I see it is possible to apply the squeeze theorem to the negative term in the denominator, but I do not know about if this is the right path. Any hint?