If \(\lim_{n\to\infty}\frac{4n+1}{n}=L\), what is \(L\)?

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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.

More practice problems: uniqueness of limits

• ZW-UNIQUENESS-OF-LIMITS-001 If a_n→2 and also a_n→ M, what is M?
• ZW-UNIQUENESS-OF-LIMITS-003 To rule out a_n→3 and a_n→7 holding together, one takes ε=(|L-M|)/(2). What is it for L=3, M=7?
• ZW-UNIQUENESS-OF-LIMITS-004 In the proof, N_1 works for convergence to L and N_2 for convergence to M. If N_1=12 and N_2=30, wh…
• ZW-UNIQUENESS-OF-LIMITS-005 Which statement is true?
• ZW-UNIQUENESS-OF-LIMITS-006 If a_n→ L and a_n→ M, then a_n-a_n=0 converges to L-M and also to 0. What follows?
• ZW-UNIQUENESS-OF-LIMITS-007 With |L-M|=0.6 and ε=0.3, for n≥ N the triangle inequality gives |L-M|≤|L-a_n|+|a_n-M|< ...
• 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…

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