Question

what is the limit of ntan(1/n) as n approaches infinity?

Answer 1

$\displaystyle \lim_{n\to \infty} n \tan \dfrac{1}{n}=\\\\ \lim_{n\to \infty} \dfrac{ \tan \dfrac{1}{n}}{\dfrac{1}{n}}=\\\\ \lim_{x\to 0} \dfrac{ \tan x}{x}=\\\\ \lim_{x\to 0} \dfrac{ \dfrac{\sin x}{\cos x}}{x}=\\\\ \lim_{x\to 0} \dfrac{\sin x }{x \cos x}=\\\\ \lim_{x\to 0} \left(\dfrac{\sin x}{x}\cdot\dfrac{1}{ \cos x}\right)=\\\\ 1\cdot1=\\ 1$