Finding limit of $\sin(x^2/2)/\sqrt{2}\sin^2(x/2)$ as $x\rightarrow0$ Can anybody help me find the limit as $x$ tends to $0$, for $$\frac{\sin(x^2/2)}{\sqrt{2}\sin^2(x/2)}.$$ How can I simplify the expression or use equivalent transformations to find a limit (without using L'Hospital)?

As $\dfrac{\sin x}x=1$, in multiplicative expressions you can replace $\sin x$ by $x$.
Hence
$$\lim_{x\to0}\frac{\sin\left(\dfrac{x^2}2\right)}{\sqrt2\,\sin^2\left(\dfrac x2\right)}
=\lim_{x\to0}\frac{\dfrac{x^2}2}{\sqrt2\,\left(\dfrac x2\right)^2}=\sqrt2.$$
