When is it allowed to "take apart" a limit?

Here's an example to show what I mean:

$\displaystyle\lim_{n\to\infty}\frac{\frac 1 ne^{\frac 1n}(e-1)}{e^{\frac 1 n}-1}=e-1$ since we can "take apart" part of the limit and find it first:

$\displaystyle\lim_{n\to\infty}\frac{\frac 1 n}{e^{\frac 1 n}-1}\overset{x=\frac 1 n}=\lim_{x\to 0}\frac x{e^x-1}\overset{LHR}=\lim_{x\to 0}\frac 1 {e^x}=1$.

It wouldn't work if we had applied LHR to the original limit.

So how do we know when it's allowed to do this or how can we know that it won't change the original limit?


If each one of the individual limits exists. Here, you're just pulling out a constant $e-1$ which you can do so long as the constant is not $0$.

For example: $$\lim_{n\to\infty} (-1)^n \frac{1}{n}\not= \lim_{n\to\infty} (-1)^n\lim_{n\to\infty}\frac{1}{n}$$ Because the right side doesn't exist.

Or: $$\lim_{n\to\infty}0*n\not=0*\lim_{n\to\infty}n$$

  • $\begingroup$ So it's alright as long as the part is not zero or infinity? $\endgroup$ – shinzou May 5 '15 at 18:42
  • $\begingroup$ It's okay as long as everything exists. If all limits exist, you can do it. $\endgroup$ – user223391 May 5 '15 at 18:43

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