Find the Lower and Upper Limits of a Sequence

The lower and upper limits of sequences have the advantage of always existing, whereas sequence limits only exist in exceptional cases. Algorithms and new functions have been added that make it possible, for the first time, to compute the lower and upper limits of oscillatory sequences as illustrated in the following.

Find the lower limit for a sequence.

Find the upper limit for a sequence.

The limit of the following sequence exists.

Hence, it is equal to both the upper and the lower limits for the sequence.

The limit of the following sequence does not exist.

This is because the even and odd subsequences converge to different limits.

Hence, the lower and upper limits of the sequence are not equal.