# some general analysis

• Oct 3rd 2006, 12:53 PM
bobby77
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
• Oct 3rd 2006, 01:05 PM
ThePerfectHacker
Quote:

Originally Posted by bobby77
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.
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2.A sequence {an} that has no limit diverges
No consider the classic example (-1)^n
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3.the sequence {1,-1,1,-1..}has two limits.
It has no limit.
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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.
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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.