<s>[INST] How to calculate the limit of $\frac{\sin(ax)}{x}$ for $x\to0$ How can I calculate the following limit epsilon-delta definition?
$$\lim_{x \to 0} \left(\frac{\sin(ax)}{x}\right)$$
Edited the equation, sorry...
 [/INST] This is easy if you know the power series of the sine function.
$$\sin(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} +\cdots$$
Then $$\sin(ax) = a x - \frac{a^3 x^3}{6} + \frac{a^5 x^5}{120} -\cdots$$
and $\frac{\sin(ax)}{x} = a - \frac{a^3 }{6}x^2 + \frac{a^5 }{120}x^4 -\cdots$ for all $x \neq 0$. Since power series are continuous you can find the limit for $x \rightarrow 0$ by simply setting $x=0$ in this expression, so the limit is $a$.
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