Properties of Limits

1.The limit, if it exists, is unique.

2.If a sequence, an, has a limit, all the subsequences have the same limit as an.

3. All convergent sequences are bounded.

4.There are bounded sequences that are not convergent.

Bounded Sequences

5. All the monotone and bounded sequences are convergent.

6.There are convergent sequences that are not monotonous.

Convergent Sequences

Rules of Limits

lim (an + bn) = lim (an) + lim (bn)

lim (an − bn) = lim (an) − lim (bn)

lim (an · bn) = lim (an) · lim (bn)

lim (an : bn) = lim (an) : lim (bn)

Rules of Limits

lim k · an =k · lim an

lim ank = (lim an)k

lim loga an = loga lim an


By applying these rules, the following cases are presented:

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits

Properties of Limits