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Does #a_n={(3/n)^(1/n)} #converge? If so what is the limit?

Answer 1

Emma Aldridge

It converges to #1#.

#f(x) = (3/x)^(1/x) = e^(ln(3/x)/x)#

#lim_(xrarroo)(ln(3/x)/x)# has initial form #(-oo)/oo# which is indeterminate, so use l'Hospital's rule.

#lim_(xrarroo) (x/3(-3/x^2))/1 = lim_(xrarr0)-1/x = 0#

Since the exponent goes to #0#, we have

#lim_(xrarroo)(ln(3/x)/x) = e^0 = 1#

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Answer 2

Ezra Adams

Yes, the sequence ( a_n = \left(\frac{3}{n}\right)^{\frac{1}{n}} ) converges. The limit of this sequence as ( n ) approaches infinity is ( 1 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.