Trouble with basic sequence limit I am having a bit of trouble with attempting to disprove that if $\lim \left(a_n - b_n\right) =0$ then $\lim \left(\frac{a_n}{b_n} \right) = 1$ (where $n \rightarrow \infty$, in all above).
I can easily disprove it if we are talking about $\lim a_n = \lim b_n$, because the limit might not exist. I.E. $a_n = (-1)^n +\frac{1}{n} $ and $b_n = (-1)^n$. But how about the limit of the division $\frac{a_n}{b_n}$? I do have a feeling it is still can be disproved, but I am not completely sure.
Also, I'm curious for the other direction. Does having  $\lim \left(\frac{a_n}{b_n} \right) = 1$ means that necessarily  $\lim \left(a_n - b_n \right)=0$ ?
 A: What about
$$a_n=\frac1n \quad b_n=\frac1{n^2}$$
and for the second direction
$$a_n=n^2+2n+1 \quad b_n=n^2$$
A: The best way to deal with such questions (and doubts) is to apply limit laws.
Let's first suppose that $$\lim_{n\to \infty} (a_n-b_n)=0$$ Assuming that $b_n\neq 0$ as $n\to\infty $ we can write the above equation as $$\lim_{n\to\infty} b_n\left(\frac{a_n} {b_n} - 1\right) =0$$ If one further assumes that $b_n$ is bounded away from $0$ ie there is a positive number $K$ such that $|b_n|>K$ as $n\to\infty$ then the above equation leads us to $$\lim_{n\to\infty} \frac{a_n} {b_n} =1$$ (Why? Because $|1/b_n|<1/K$ so that $1/b_n$ is bounded and multiplying the above equation with $1/b_n$ does the job.)
But this is only a sufficient condition. If $b_n$ is not bounded away from $0$ then we need to analyze the ratio $a_n/b_n$ directly and we can't infer anything from $a_n-b_n\to 0$.

We can work in reverse and start with the given assumption that $a_n/b_n\to 1$ and try to infer some information about limiting behavior of $a_n-b_n$. Just like before we have $$a_n-b_n=b_n\left(\frac{a_n} {b_n} - 1\right)$$ Here one must ensure that $b_n$ must not become too large to offset the fact that $(a_n/b_n) - 1\to 0$. In particular if $b_n$ is bounded then we have $a_n-b_n\to 0$. But nothing can be inferred if $b_n$ is unbounded.
Both the above treatments are based on the following simple result:

Theorem: If $|a_n| $ is bounded and $b_n\to 0$ then $a_nb_n\to 0$.

You can prove the above theorem easily using Squeeze Theorem.

Moral of the story: do not treat the limit of a sequence like the terms of the sequence. Thus provided $b_n\neq 0$ we have $$a_n-b_n=0\iff \frac{a_n} {b_n} =1$$ but the same implications don't hold if we also use the limit operation. The limit operation does follow laws like algebra of numbers, but with certain restrictions and in general one must be careful to take note of those restrictions. 
A: Let $a_n=2b_n$ and $\lim a_n=0$ (for example $a_n={1\over n}$ or anything like that).
For the second one, take $$a_n=b_n+c_n$$where $$\lim c_n\ne 0$$and $$\lim{c_n\over b_n}=0$$i.e. $c_n=o(b_n)$ for example $$c_n=1\\b_n=n$$therefore $$\lim a_n-b_n=\lim c_n\ne 0$$
