Limits

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Uziha

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Hello, would you please help me calculate the limit of ((1/n)-1)^n Thanks
 
Uziha said:
Hello, would you please help me calculate the limit of ((1/n)-1)^n Thanks
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[imath]\left[\dfrac{1}{n}-1\right]^n[/imath] you did not say what [imath]n\to~?[/imath]
 
Uziha said:
Hi it is infinity
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Now please show us what you have tried, so we can see what methods you are using, and where you need help.

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I think that ((1/n)-1)^n = ((1/n)-1)^1 * ((1/n)-1)^(n-1) = -1 * e = -e. Because (n-1) seems to me like moving the series of numbers. Thank you.
 
Let y = [imath]\left[\dfrac{1}{n}-1\right]^n[/imath]

Then ln y = [imath]n\ln\left[\dfrac{1}{n}-1\right][/imath]
Now compute the limit of both sides

Can you also confirm that you posted the correct problem?
 
Uziha said:
I think that ((1/n)-1)^n = ((1/n)-1)^1 * ((1/n)-1)^(n-1) = -1 * e = -e. Because (n-1) seems to me like moving the series of numbers. Thank you.
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Have you noticed that (1/n)-1 is a negative number, so that raising it to the nth power results in alternating signs?

What does that tell you about the limit?

This suggests either that the problem was intended to be something different, as @Jomo has hinted, or that you are missing the obvious (which will become all the more obvious if you follow Jomo's suggestion!). You noticed the -1; that should have been a warning sign.
 
I think I understand. If so, the numbers 1 and -1 are the limits.
 
Uziha said:
I think I understand. If so, the numbers 1 and -1 are the limits.
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Are you familiar with the epsilon-delta definition of a limit? A limit is what the functional value approaches--it gets closer and closer to that value=limit. Does this align with your answer?
 
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Yes I am familiar with the definition, and according to what was written, now I can understand that there are no limits, but there are partial limits with the number -e and e. Thank you.
 
Can you define partial limits as I am unfamiliar with that term?
 
Jomo said:
Can you define partial limits as I am unfamiliar with that term?
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Uziha said:
Partial means 2 subsequences of the original sequence
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That is not really a mathematical definition. But the term has come up here before, and I found then that it is used occasionally. Here is one book containing a definition. The same definition is given on page 5 here, with a definition of the word "frequently" as used in it. A clearer definition is found on page 22 here.
 
L is a partial limit of (xn) if for every neighborhood U of L, (xn) is frequently in U.
What the heck does frequently mean?!

The clearer definition from Dr Peterson's link is much clearer than the other definition.

To OP, I am sorry thinking that the phrase partial limit does not exist. It's not common at all and it is the first time I ever heard of it.
 
Jomo said:
To OP, I am sorry thinking that the phrase partial limit does not exist. It's not common at all and it is the first time I ever heard of it.
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Actually, you've seen it at least once before:

Give an example to an unbounded sequence that the set of all its real partial limits is bounded

1. Give an example of an unbounded sequence that the set of all its real partial limits is bounded. 2. Give an example of an unbounded sequence that the set of all its read partial limits is unbounded
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Of course, I only found that by searching for the term; I didn't recall this page explicitly! And the fact that it was on the first page of hits suggests how rare the term is!

Jomo said:
L is a partial limit of (xn) if for every neighborhood U of L, (xn) is frequently in U.
What the heck does frequently mean?!
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As I mentioned, my second like includes a definition of "frequently", prior to using it! Presumably the book had also done so earlier, but I didn't look.
 
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