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 $\lim \Big(\dfrac {x_n}{y_n}\Big)=1$ ?


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.