If $(x_n)$ , $(y_n)$ are sequences of non-zero real numbers such that $\lim (x_n-y_n)=0$ , then is it true
that $\lim \Big(\dfrac {x_n}{y_n}\Big)=1$ ?
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No. Take $x_n = 1/n$ and $y_n=1/n^2$.
On the other hand, if you require in addition that $\liminf y_n = \delta > 0$ (or almost equivalently $|y_n| > \delta$ for all $n > 0$), then you can write
$$\lim \frac{x_n}{y_n} = \lim \frac{y_n + (x_n-y_n)}{y_n} = \lim \left( 1 + \frac{x_n-y_n}{y_n} \right),$$ where $\limsup |(x_n-y_n)/y_n| \leq \limsup |x_n-y_n|/\delta = 0$, so that the limit is in fact, one.
