Monday, January 18, 2016

calculus - Another proof of limlimitsthetato0fracsinthetatheta=1



I am sorry if this is a duplicate question, but as far as I searched I have not come across this question.





lim. This formula is famously proven by geometrical means using area of a circle and so on..




I want to know if this method of proof over l-hopital's Rule is also acceptable.
\lim\limits_{\theta \to 0} \frac{\sin\theta}{\theta} = \lim_{\theta \to 0} \frac{\cos \theta}{1} = 1
If there is any other method to prove apart from the two methods above. Please Share. Thank You!


Answer



sin(\theta)=\sum_{n=0}^\infty \frac{(-1)^n{\theta}^{2n+1}}{(2n+1)!}=\theta-\frac{(\theta)^3}{3!}+\frac{(\theta)^5}{5!}+....then\frac{sin(\theta)}{\theta}=1-\frac{(\theta)^2}{3!}+\frac{(\theta)^4}{5!}+.... and the limit as \theta\rightarrow 0 is 1


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...