Find $\lim_{x\to0}{\frac{\arctan (2x)}{3x}}$ without using '0/0=1' Find limit as 
$$
\lim_{x \to 0} \frac{\arctan (2x)}{3x}
$$
without using $\frac{0}{0} = 1$.
I wanted to use 
$$
\frac{2}{3} \cdot \frac{0}{0} = \frac{2}{3} \cdot 1,
$$ 
but our teacher considers $\frac{0}{0}$ a "dangerous case" and we are not allowed to use this method.
We have not studied integrals yet, so I cannot use integral formulas. We have only studied a bit of derivative taking...
I tried many substitutions, but failed :( Any suggestion?

You may use L'Hospital's rule:
$$
\lim_{x\to0}{\frac{\arctan (2x)}{3x}}=\lim_{x\to0}{\frac{\frac{2}{1+4x^2}}{3}}=\frac23.
$$
