lim x->o (tan( x/3)/x) ???

Can someone help... $\lim x \to o \left(\tan \frac{\frac{x}{3}}{x}\right)$

Jun 20, 2018

We have to use that ${\lim}_{x \to 0} \sin \frac{x}{x} = 1$ to solve the indetermination

Explanation:

let's manipulate the expression a little bit first:

${\lim}_{x \to 0} \left(\tan \frac{\frac{x}{3}}{x}\right) = {\lim}_{x \to 0} \left(\frac{\sin \frac{\frac{x}{3}}{\cos} \left(\frac{x}{3}\right)}{x}\right) = {\lim}_{x \to 0} \left(\sin \frac{\frac{x}{3}}{\cos \left(\frac{x}{3}\right) x}\right)$ and now dividing by 3

${\lim}_{x \to 0} \left(\tan \frac{\frac{x}{3}}{x}\right) = {\lim}_{x \to 0} \left(\frac{\sin \frac{\frac{x}{3}}{3}}{\cos \left(\frac{x}{3}\right) \left(\frac{x}{3}\right)}\right) = {\lim}_{x \to 0} \left(\left(\sin \frac{\frac{x}{3}}{\frac{x}{3}}\right) \cdot \frac{1}{\cos \left(\frac{x}{3}\right) \cdot 3}\right)$

But we know that:

${\lim}_{x \to 0} \left(\sin \frac{\frac{x}{3}}{\frac{x}{3}}\right) = 1$, and that

${\lim}_{x \to 0} \frac{1}{\cos} \left(\frac{x}{3}\right) = 1$, so the limit is

${\lim}_{x \to 0} \left(\tan \frac{\frac{x}{3}}{x}\right) = {\lim}_{x \to 0} \left(\frac{\sin \frac{\frac{x}{3}}{3}}{\cos \left(\frac{x}{3}\right) \left(\frac{x}{3}\right)}\right) = {\lim}_{x \to 0} \left(\left(\sin \frac{\frac{x}{3}}{\frac{x}{3}}\right) \cdot \frac{1}{\cos \left(\frac{x}{3}\right) \cdot 3}\right) = 1 \cdot 1 \cdot \frac{1}{3} = \frac{1}{3}$