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Could someone define what it means for a limit to "exist"?

Must the limit after using L'Hospital Rule approach a specific value?

What if the limit after using L'Hospital Rule approaches infinity? Does it count as being existent?

i.e.

$$\lim_{n\rightarrow\infty} \frac{\sqrt{n}}{(\log(n))^2}$$

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It's correct if the limit is infinity. The limit must exists, no matter if it is finite or not. On the other hand if you use l'Hopital rule and find that the limit does not exist you cannot conclude that the initial limit does not exist. In that case you must use another methods to analyse the limit.

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  • $\begingroup$ What does it mean for a limit to not exist? I know that if the limit oscillates between 2 values then it doesn't exist. Any other cases that would qualify a limit to "not exist"? $\endgroup$ – A_for_ Abacus Oct 25 '14 at 22:15
  • $\begingroup$ Also, just to clarify on what you said, when a limit equals infinity (not a finite value). The limit still counts as being existent? $\endgroup$ – A_for_ Abacus Oct 25 '14 at 22:24
  • $\begingroup$ If after applying l'Hôpital's rule you find a non existent limit, you can't conclude that the given limit doesn't exist. $\endgroup$ – egreg Oct 25 '14 at 22:50
  • $\begingroup$ When the limit is infinite the limit exists. The limit does not exist when you can't find a number L the function approaches to that number for a particular value of x. Take for instance sin(x) function which does not have limit when x approaches infinity. That is beaucase there is not such a number L. For a formal proof search on google. $\endgroup$ – SebiSebi Oct 26 '14 at 10:49

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