Why does a table of values not prove a limit?

Question

Why does the method of a table of values only provide a reasonable guess and not proof of the limit value?

thanks!!

Answer 1

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A table of values of f(x) around x = a does just that: it gives a list of values around that x = a. In order to show prove the value of lim (x-->a) f(x), we can get as close as we want to its value by restricting x to a sufficiently small neighborhood around x = a (this gives rise to the epslion-delta definition of a limit). A table of values around x = a gives us several of these neighborhoods, but in order to prove the value of the limit, we need to check all possible neighborhoods, something that a table of values cannot do.

I hope this helps!

Answer 2

Proofs require the definition.