Calculate the maximum of $f(x,y)=\left|\frac{\sin(xy)}{x\sqrt{y}}\right|$

Question

Calculate the maximum of $$f(x,y)=\left|\frac{\sin(xy)}{x\sqrt{y}}\right|\, , \quad \text{for} \ x\in\mathbb R\, , \, y\in\mathbb R^{+}\, .$$ I suspect that this function is unbounded. In fact: $$f(x,y)=\left|\frac{\sin(xy)}{x\sqrt{y}}\right|= \left|\frac{\sin(xy)}{xy}\right| \sqrt{y}$$ but $\left|\frac{\sin(xy)}{xy}\right|\leq 1$ and $\sqrt y\to \infty$. In particular, fixed $x\in\mathbb R$ if we consider (for example) a sequence $y_k(x)= \frac{\frac{\pi}{2}+k\pi}{x}$ we have that $\lim_{k\to \infty} y_k(x)=\infty$ and $$\lim_{y_k\to \infty} f(x,y_k)=\infty\, .$$

Answer 1

It is not true that the limi is $\infty$ for every $x$. But if you take, say, $x=1$, then what you did is correct.