Suppose that $a_n$ and $b_n$ have a finite limit. Then is it true that $\lim_{n \rightarrow\infty} \frac{b_n}{a_n}=1$ is enough to ensure that $\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=1$?
Attempt:
If $\lim_{n \rightarrow\infty} \frac{b_n}{a_n}=1$ then $\lim_{n \rightarrow\infty}\frac{1}{ \frac{b_n}{a_n}}=\frac{1}{1}=1$
If $x_n = b_n/a_n \rightarrow L \neq 0$, then $x_n^{-1} = a_n/b_n \rightarrow L^{-1}.$
We have
$$|x_n^{-1}- L^{-1}| = \frac{|x_n - L|}{|x_n||L|}.$$
If $(x_n)$ converges then it is bounded and $|x_n| > K > 0$ for $n$ sufficently large if the limit is non-zero.
Note that $||x_n| - |L|| \leq |x_n - L|< |L|/2$ for $n$ sufficiently large and we can take $K = |L|/2$.
Hence
$$|x_n^{-1}- L^{-1}| < \epsilon$$ when
$$|x_n - L| < K|L|\epsilon.$$
which will be true for all $n$ sufficiently large.
In this case $L = 1$.
That works quite well. When limits of sequences are finite and nonzero, the limit of a quotient is the quotient of the respective limits. To wit, let $x_n = 1 \forall n$ and $y_n = a_n / b_n$. Then $$\lim_n x_n / y_n = \lim_n x_n / \lim_n y_n = 1/1 = 1$$
which, as you noted, is what we wanted in the first place.
