Limits to infinity - some wonders $a_n \to 0$ , $b_n \to \infty$ Is it true that: A) $$\lim \limits_{n \to \infty}(a_n - b_n) = -\infty$$ B)$$\lim \limits_{n \to \infty}\frac{a_n}{b_n}=0$$ Well about A I couldn't find a counter example so I'm tending to say that A is true. About B I think its true too but I know that $\infty $ limits tend to be tricker than the human logic. I also curios what will happen if I do: $$\lim \limits_{n \to \infty}\frac{b_n}{a_n}$$ Will the limit in this case be $\infty$?

$A$ and $B$ are true by arithmetic of limits:

sum of limits acts like you would expect as far it's not "$\infty-\infty$", and for B consider $|a_n/b_n|\le |1/b_n|$ starting at some $N$

$C$ is false, you could get for example limit $-\infty$: $a_n=-1/n$, $b_n=n$.

while: $b_n/a_n=-n^2\rightarrow -\infty $