Wednesday, May 23, 2018

limits - Evaluating and proving $limlimits_{xtoinfty}frac{sin x}x$

I've just started learning about limits. Why can we say $$ \lim_{x\rightarrow \infty} \frac{\sin x}{x} = 0 $$ even though $\lim_{x\rightarrow \infty} \sin x$ does not exist?



It seems like the fact that sin is bounded could cause this, but I'd like to see it algebraically.



$$ \lim_{x\rightarrow \infty} \frac{\sin x}{x} =
\frac{\lim_{x\rightarrow \infty} \sin x} {\lim_{x\rightarrow \infty} x}
= ? $$




L'Hopital's rule gives a fraction whose numerator doesn't converge. What is a simple way to proceed here?

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...