Question

Examine if the following student definitions clearly capture all sequences that have limits and clearly exclude all sequences that don’t have limits. Explain how you can tell.

(a) Amy’s definition L is a limit of a sequence {an}∞ n=1 if and only if an approaches L as n approaches ∞.

(b) Brittany’s definition L is a limit of a sequence {an}∞ n=1 if and only if an gets closer to but never reaches L as n approaches ∞.

(c) Cindy’s definition L is a limit of a sequence {an}∞ n=1 if and only if the difference between an and L gets smaller as n approaches ∞.

(d) Dorothy’s definition L is a limit of a sequence {an}∞ n=1 if and only if the difference between

Answer 1

All the definition of all the students are correct. The meaning of all the definition is same. We can find out the limit by using these definition. These definition clearly captured the all sequences that have limits and clearly exclude all sequences that don’t have limits.

For eg.

let us take an= (1/n) so as n approaches to ∞ the (1/n ) tends toward zero. Or the difference between an and zero is becoming less and less.

Therefore all above definition are correct