Limit that fail to exist

Question

Does a limit that equals to infinity considered to exist ?? am confused !!

for Example 1/(x-2)--> when evaluating the limit at 2 the result is 1/0 which is infinity while after looking at the graph the left hand limit equals -inf while right hand limit equals +inf does the one sided limits exists ??

pls an expert discuss existence of limits for my question and example because am confused

thanx

Answer 1

The limit exists if the values approach some real number. If the values are getting bigger and bigger the limit doesn't exist. There are other ways the limit could not exist. Saying the limit is infinity is being more specific about how the limit fails to exist.

Also, saying "the result is 1/0" causes many math folks pain. You could say the denominator is going to zero, so the whole thing is going to infinity. Or if you don't want to say that much, just say the limit is infinity.

Answer 2

If we consider the real number or complex number system then the limit does not exist. But if we include ∞ with that system, which is called extended real number or extended complex number system, then obviously we have to say that the limit exists.