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To check for the existence of a limit of a function at a point, you can use the following conditions:
The function must be defined in a punctured neighborhood of the point.
The limit of the function as it approaches the point must exist and be finite.
How do you know a limit does not exist? In short, the limit does not exist if there is a lack of continuity in the neighbourhood about the value of interest. Recall that there doesn't need to be continuity at the value of interest, just the neighbourhood is required.