Difference Between A Limit Going To ±Infinity and A Limit That Doesn't Exist
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Many people get confused between the case when a limit goes to infinity (or negative infinity) and when a limit does not exist. And there is a good reason for the confusion. Many discussions I've read do not separate the two cases. But they are different.
When a Limit DOES Exist
When a limit goes to positive or negative infinity, the limit DOES exist. The limit is exactly that, positive or negative infinity.
When a Limit DOES NOT Exist
There is only one case when a limit doesn't exist: when the limit is different from the left than it is from the right. This concept requires understanding one-sided limits. There are videos on that page showing examples of when the limit doesn't exist.
For more detail including graphs, see the substitution section on the finite limits page. And, as usual, check with your instructor to see how they define limits that do not exist.
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