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Math Help - Limit Proof

  1. #1
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    Limit Proof

    What would be the easiest way to prove that the lim (as n approaches infinity) of (2)/(n+1)=0.

    Thanks.
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  2. #2
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    This one is easy: 2/[n+1] < 2/n.
    If e>0 then there is a positive integer K such that (1/K)<(e/2) or (2/K)<e.
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  3. #3
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    Quote Originally Posted by JaysFan31 View Post
    What would be the easiest way to prove that the lim (as n approaches infinity) of (2)/(n+1)=0.

    Thanks.
    Note the sequnce,
    a_n={2/(n+1)}
    Is identical to the sequence,
    b_n={2(1/n)/(1+1/n)}

    Now the numerator is,
    lim 2(1/n)
    Since, lim (1/n)--->0
    So too, lim 2(1/n)---0 by constant function rule.

    Next, the demonator is,
    lim (1+1/n)
    But, lim 1=1 and lim (1/n)=0
    Thus by the rule of seuqnces addition,
    lim (1+1/n) exists and is, 1+0=1 (not equal to zero)
    Thus, by the rule of sequnces division,
    lim 2(1/n)/(1+1/n)--->0
    Thus,
    lim a_n--->0
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