Evaluate $\lim_{x\to+\infty} \frac{\sqrt{x}(\sin{x}+\sqrt{x}\cos{x})}{x\sqrt{x}-\sin({x\sqrt{x})}}$

Question

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?

Answer 1

HINT

You should factor $x\sqrt{x}$ instead of $\sqrt{x}$. Precisely, \begin{align*} \frac{\sqrt{x}(\sin(x) + \sqrt{x}\cos(x))}{x\sqrt{x} - \sin(x\sqrt{x})} = \frac{\frac{\sin(x)}{x} + \frac{\cos(x)}{\sqrt{x}}}{1 - \frac{\sin(x\sqrt{x})}{x\sqrt{x}}} \end{align*}

Can you take it from here?