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Let $\lim = \lim_{n \to +\infty}$.

When is $\lim \frac{f(n)}{g(n)} = \frac{\lim f(n)}{\lim g(n)}$ always ?

I know that $\lim g(n) \neq 0$ is one condition. Are there any other conditions?

Also, when is $\lim \left [g(n) - f(n) \right ] = \lim f(n) - \lim g(n)$ always? Like I know there is a problem when $\lim \left [g(n) - f(n) \right ]$ is finite, but the individual limits are not.

Can anyone summarize the conditions?

rims
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    The limits of $f$ and of $g$ must actually exist. The limit of one of them must be finite. If both are infinite you have trouble. If both are finite all is good (unless the denom is $0$). If one is finite that the other infinite, I think things are okay if you are using the terminology that allow you to say "the limit equal infinity" (Many text won't allow that.) – fleablood Oct 16 '19 at 01:22
  • It is best to express it the form of a product. If $\lim f$ exists and is non zero then the limiting behavior of $fg$ is same as that of $g$. For quotient write $f/g=f(1/g)$ – Paramanand Singh Oct 16 '19 at 01:36
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    See this thread for more details. https://math.stackexchange.com/questions/2971122/extended-limit-laws-for-step-by-step-evaluation-of-limits – Paramanand Singh Oct 16 '19 at 01:42

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