Exercise · ZW-UNIQUENESS-OF-LIMITS-003 · ★☆☆ Basic
To rule out \(a_n\to3\) and \(a_n\to7\) holding together, one takes \(\varepsilon=\frac{|L-M|}{2}\). What is it for \(L=3\), \(M=7\)?
Choose one option.
Show answer
- C. \(2\)
Explanation
\(\varepsilon=\frac{|3-7|}{2}=\frac42=2\).
Review the concept: Uniqueness of limits
Uniqueness of limits says that a sequence \((a_n)\) cannot converge to two different numbers \(L\) and \(M\); that is why it makes sense to speak of “the” limit.
\[a_n\to L,\ a_n\to M\ \Rightarrow\ L=M\]
More practice problems: uniqueness of limits
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- ZW-UNIQUENESS-OF-LIMITS-008 An alternative proof shows |L-M|<ε for every ε>0. Which fact lets us conclude L=M?
- ZW-UNIQUENESS-OF-LIMITS-009 If (-1)^n→ L, with ε=tfrac12 the terms 1 and -1 (which appear for large n) satisfy |1-L|<tfrac12 an…