Saturday, October 6, 2018

calculus - Compute limnto+inftynleft(tanleft(fracpi3+frac1nright)sqrt3right) without using L' Hôpital


Compute lim without using L'Hospital's rule.





By using L'Hospital's rule and



\tan'( \Diamond )=( \Diamond )'(1+\tan^{2}( \Diamond ))
I mean by \Diamond a function
so I got
\begin{align} \lim_{n\to +\infty}n\left(\tan\left(\dfrac{\pi}{3}+\dfrac{1}{n} \right)-\sqrt{3}\right) &=\lim_{n\to +\infty}\dfrac{\left(\tan\left(\dfrac{\pi}{3}+\dfrac{1}{n} \right)-\sqrt{3}\right)}{\dfrac{1}{n}}\\ &=1+\tan^{2}\left(\dfrac{\pi}{3}\right)=1+\sqrt{3}^{2}=1+3=4 \end{align}



I'm interested in more ways of computing limit for this sequence.

No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f \colon A \rightarrow B and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...