limit question

**cinder** · February 22nd 2006, 05:57 PM

$\begin{array}{cc}\lim\\x\rightarrow0\end{array} \frac{\sin2x}{\sin3x}$

How can I solve that without using L'Hôpital's rule (which is considered illegal in my calculus class)?

**TD!** · February 23rd 2006, 10:57 AM

For small x (which we have here, since we're letting x tend to 0) you can use the first order Taylor expansion for sin(x) which is x. So for small x:

$\frac{{\sin \left( {2x} \right)}}{{\sin \left( {3x} \right)}} \approx \frac{{2x}}{{3x}} \Rightarrow \mathop {\lim }\limits_{x \to 0} \frac{{\sin \left( {2x} \right)}}{{\sin \left( {3x} \right)}} = \mathop {\lim }\limits_{x \to 0} \frac{{2x}}{{3x}} = \frac{2}{3}$

It's also possible that you have to use the 'standard limit':

$\mathop {\lim }\limits_{\alpha \to 0} \frac{{\sin \alpha }}{\alpha }=1$

In that case, rewrite as follows:

$\mathop {\lim }\limits_{x \to 0} \frac{{2\sin \left( {2x} \right)}}{{2x}}\frac{{3x}}{{3\sin \left( {3x} \right)}} = 2\mathop {\lim }\limits_{x \to 0} \frac{{\sin \left( {2x} \right)}}{{2x}} \cdot \frac{1}{3}\mathop {\lim }\limits_{x \to 0} \frac{{3x}}{{\sin \left( {3x} \right)}} = \frac{2}{3}$

**ThePerfectHacker** · February 23rd 2006, 12:43 PM

Cinder I realize that this is off topic. I looked at your LaTex code an saw that you wrote that limit in array mode. I think it is easier if you write it as:
\lim_{n\to 0}\frac{\sin 2x}{\sin 3x}

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