Sequential Criterion for Limits The sequential criterion is a formal definition for determining whether a limit exists. It establishes a necessary and su...
The sequential criterion is a formal definition for determining whether a limit exists. It establishes a necessary and sufficient condition for a limit to exist.
Definition:
Let (f) be a function defined on an open interval (I), and let (L) be a real number. We say that (\lim_{x\to a} f(x) = L) if for every (\epsilon>0), there exists a (\delta>0) such that if (0<|x-a|<\delta), then |f(x)-L|<\epsilon).
Intuitive Interpretation:
The sequential criterion says that the limit is equal to (L) if, for any given (\epsilon>0), we can find a (\delta>0) such that the difference between (f(x)) and (L) is smaller than (\epsilon) for all (x) within the interval (I).
Examples:
If (f(x) = x^2) and (a=0), then (\lim_{x\to 0} f(x) = L) because for any ( \epsilon>0), we can find (\delta>0) such that if (0<|x|<\delta), then (|x^2 - 0| = |x|<\epsilon).
If (f(x) = \begin{cases}
1 & x\ge 0 \\
0 & x\le 0
\end{cases}), then (\lim_{x\to 0} f(x) = 1) because for any ( \epsilon>0), we can find (\delta>0) such that if (0<|x|<\delta), then |1-0| = |x|<\epsilon).
Implications of the Sequential Criterion:
The sequential criterion is necessary for a limit to exist. This means that if the limit exists according to the sequential criterion, it must equal the actual limit.
The sequential criterion is also sufficient for a limit to exist. This means that if a limit exists according to the sequential criterion, it must exist according to the standard definition.
Note:
The sequential criterion is a powerful tool for determining limits, but it is important to note that it has some limitations. For example, the sequential criterion does not work for all functions, especially those with jumps or breaks