Lim 3x/tan3x x →0 How to solve it ? I think the answer will be 1 or -1 who can solve it ?

1 Answer
Mar 7, 2018

The limit is $1$.

Explanation:

$L i {m}_{x \to 0} \frac{3 x}{\tan 3 x}$

=$L i {m}_{x \to 0} \frac{3 x}{\frac{\sin 3 x}{\cos 3 x}}$

=$L i {m}_{x \to 0} \frac{3 x \cos 3 x}{\sin 3 x}$

=$L i {m}_{x \to 0} \frac{3 x}{\sin 3 x} . \cos 3 x$

=$L i {m}_{x \to 0} \textcolor{red}{\frac{3 x}{\sin 3 x}} . \cos 3 x$

=$L i {m}_{x \to 0} \cos 3 x$

=$L i {m}_{x \to 0} \cos \left(3 \cdot 0\right)$

$= C o s \left(0\right) = 1$

Remember that:

$L i {m}_{x \to 0} \textcolor{red}{\frac{3 x}{\sin 3 x}} = 1$

and

$L i {m}_{x \to 0} \textcolor{red}{\frac{\sin 3 x}{3 x}} = 1$