Question

Evaluate
\[\lim_{\theta \rightarrow 0} \frac{tan \theta}{\theta ^2 cot3\theta}\]

Answer 1

use lim theta--->0 tan(theta)/theta=1

Answer 2

Oh.. wait.. I thought I made some mistakes in previous calculation!

Answer 3

lim theta--->0 tan(thata) tan (3theta)/ theta^2

Answer 4

\[=\lim_{\theta \rightarrow 0} \frac{tan \theta}{\theta ^2cot3\theta}\]\[=\lim_{\theta \rightarrow 0} \frac{sin \theta}{\theta cos \theta}\frac{sin3\theta}{\theta cos3\theta}\]\[=(1)\lim_{\theta \rightarrow 0} 3\frac{sin3\theta}{3\theta cos3\theta}\]\[=(1)(3)=3\]

Answer 5

That is it!

Answer 6

yes

Answer 7

*Note:
\[\lim_{x \rightarrow 0}\frac{sinx}{x} = 1\]\[\lim_{x \rightarrow 0}\frac{sinx}{xcosx} = \lim_{x \rightarrow 0}\frac{sinx}{x} \times \lim_{x \rightarrow 0}\frac{1}{cosx} = 1 \]So,
\[\lim_{x \rightarrow 0}\frac{tanx}{x}=1\]