Rate of convergence for sequences I have numerically determined that the sequence $\{f_x\} = \frac{\sin(x^2)}{x^2}$ approaches $1$ (as $x$ approaches $0$) faster than the sequence $\{g_x\} = \frac{\sin^2(x)}{x^2}$. However, I am stuck for determing the rate of convergence of each sequence.
Should I use the small angle approximation $\sin(x)\approx x$ and the fact that $|\sin(x)|\leq 1$? Thanks.

Use Taylor expansion:
$$
f_x=\frac{1}{x^2}\Bigl(x^2-\frac{x^6}{6}+O(x^{10})\Bigr)=1-\frac{x^4}{6}+O(x^8).
$$
$$
g_x=\frac{1}{x^2}\Bigl(x-\frac{x^3}{6}+O(x^{5})\Bigr)^2=1-\frac{x^2}{3}+O(x^4).
$$
Added detail on the expansion of $g_x$:
$$
\Bigl(x-\frac{x^3}{6}+O(x^{5})\Bigr)^2=x^2-2\cdot x\cdot\frac{x^3}{6}-2\cdot x\cdot O(x^5)+\frac{x^6}{36}+O(x^{10})=x^2-\frac{x^4}{3}+O(x^6)
$$
