How do you know if a certain limit does not exist - without actually graphing it? What answer do you get that shows you there is no such limit?
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$\begingroup$ What do you mean by "arithmetically"? How do you define the limit of a sequence/function? $\endgroup$– Brian61354270Nov 24, 2019 at 21:16
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$\begingroup$ I mean not by drawing it out. $\endgroup$– BurtNov 24, 2019 at 21:18
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$\begingroup$ Usually I know it doesn't exist, because I proved it does not exist :) $\endgroup$– Severin SchravenNov 24, 2019 at 21:21
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1$\begingroup$ I would encourage you look over any of these existing questions about this topic: math.stackexchange.com/search?q=limit+not+exist $\endgroup$– Brian61354270Nov 24, 2019 at 21:22
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1 Answer
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This is rather broad, but in general:
- In univariate calculus, if you can show that the limit from the left is not the same as the limit from the right, i.e. $\lim \limits_{x \to c^+} f(x) \ne \lim \limits_{x \to c^-}f(x)$, then it follows that the limit doesn't exist.
- In multivariable calculus, you can do the same thing but showing different values of the limit for two different paths to a certain point.
