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Math Help - some general analysis

  1. #1
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    some general analysis

    Indicate whether the following statements are true or false

    1.A sequence {an} can have more than one limit.
    2.A sequence {an} that has no limit diverges
    3.the sequence {1,-1,1,-1..}has two limits.
    4.If lim n->inf {an}=L and Lim n-> inf{bn}=M then lim n->inf {an bn } =LM

    5 f(x)=(x^2-1)/(x-1) and g(x) =x+1 are same function
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  2. #2
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    Quote Originally Posted by bobby77 View Post
    Indicate whether the following statements are true or false

    1.A sequence {an} can have more than one limit.
    No. If the limit exists it is unique. CaptainBlank gave a nice proof of this.
    2.A sequence {an} that has no limit diverges
    No consider the classic example (-1)^n
    3.the sequence {1,-1,1,-1..}has two limits.
    It has no limit.
    4.If lim n->inf {an}=L and Lim n-> inf{bn}=M then lim n->inf {an bn } =LM
    Yes. But it does not work the other way around always, so you need to be careful.
    5 f(x)=(x^2-1)/(x-1) and g(x) =x+1 are same function
    No, it can be shown two functions are equal only if they have the same domain and for each x in D we have f(x)=g(x).
    The domain of f(x) doth not include 1.
    But g(x) does.
    Equality is wrong.
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