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    (Original post by generalebriety)
    I don't see the expression you posted on that Wikipedia page anywhere. Can you type out precisely what your textbook says and/or point me to the line you're talking about on that page?
    From Wikipedia:

    The indeterminate forms include 0^0, 0/0, 1^\infty, \infty - \infty, \infty/\infty, 0 \times \infty, and \infty^0.
    I'm obviously referring to 1^\infty there.
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    Ah! Wait, I think I see what you're on about - you're on about the indeterminate form 1^\infty.

    Let me explain it by means of another example: the form 0^0, when working over the real numbers*. Provided this is defined and well behaved, we can write it in two different ways: \displaystyle \lim_{x\to 0^+} 0^x and \displaystyle \lim_{y\to 0^+} y^0. Now, for any x > 0, we have 0^x = 0, and so if we take the first limit, we see that \displaystyle \lim_{x\to 0^+} 0^x = 0; however, for any y > 0, we have y^0 = 1, and so taking the second limit, \displaystyle \lim_{y\to 0^+} y^0 = 1. This gives us two different expressions for 0^0; in order for 0^0 to be a sensible and usable quantity, we need these two limits to be equal, otherwise we don't know how we're allowed to work with 0^0 (if at all); as they're not, we say 0^0 is undefined.

    Now, 1^infinity is undefined for the same reason. While it's obvious that \displaystyle \lim_{x\to\infty} 1^x = 1, it's also fairly obvious that (if you'll excuse this somewhat informal treatment) \displaystyle \lim_{y\to 1^+} y^\infty = \infty. Or, a more rigorous one: \displaystyle \lim_{z\to\infty} (1 + \frac{1}{z})^z = e, which is slightly harder to prove (try binomial expansion, using the fact that e = 1/0! + 1/1! + 1/2! + ...). Basically, we can't assign a value to 1^infinity because it can be reached in different ways and these all give different values.

    * Though, actually, 0^0 has a very clear and well defined meaning when we're working over the natural numbers, and 0^0 = 1 there. This is because we're working in a completely different system, though - we're counting rather than measuring, and in this context, the quantity "n^k" tends to mean "the number of distinct sequences of size k made up only of values from the first n integers". Well, there's only one sequence of size 0 made up of values from the first 0 integers - the empty one. So 0^0 = 1.
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    Well I think that pretty much answers it for me, thank you very much Just one thing, I don't see what you meant with the e example: to me the main difference, and the reason why I was so perplexed about 1^\infty, is that if you plot the function f(x) = (1 + \frac{1}{x})^x you can clearly see it converging to e as x \to +\infty. On the other hand, if you plot f(x) = 1^x, it just comes out as a straight line, just like you would expect. But I guess it all sort of makes sense now
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    (Original post by Emc2)
    Well I think that pretty much answers it for me, thank you very much Just one thing, I don't see what you meant with the e example: to me the main difference, and the reason why I was so perplexed about 1^\infty, is that if you plot the function f(x) = (1 + \frac{1}{x})^x you can clearly see it converging to e as x \to +\infty. On the other hand, if you plot f(x) = 1^x, it just comes out as a straight line, just like you would expect. But I guess it all sort of makes sense now
    Well, do you not agree that the line f(x) = 1 converges to 1 as x --> infinity? (I realise that you're probably used to thinking of functions converging to limits in cases where they never quite reach that limit, but think about it in this way: if you don't agree that that function converges to 1, you're forced to agree that it has to converge to something else or it has to diverge.)
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    (Original post by generalebriety)
    Well, do you not agree that the line f(x) = 1 converges to 1 as x --> infinity? (I realise that you're probably used to thinking of functions converging to limits in cases where they never quite reach that limit, but think about it in this way: if you don't agree that that function converges to 1, you're forced to agree that it has to converge to something else or it has to diverge.)
    Yes, that was exactly my point Before you explained it to me I couldn't see why 1^\infty was indeterminate, but now I know it's because I was using the "wrong" limit to illustrate that. I guess it's just my crappy textbook's fault for not explaining it properly in the first place :rolleyes:
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    (Original post by Emc2)
    Yes, that was exactly my point Before you explained it to me I couldn't see why 1^\infty was indeterminate, but now I know it's because I was using the "wrong" limit to illustrate that. I guess it's just my crappy textbook's fault for not explaining it properly in the first place :rolleyes:
    Sorry, I'm a bit confused. You said you didn't see what I meant with one example - do you understand what's going on now or still need some clarification?
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    (Original post by generalebriety)
    Sorry, I'm a bit confused. You said you didn't see what I meant with one example - do you understand what's going on now or still need some clarification?
    I get it now I just didn't quite understand why you felt the need to bring up the e example, but all the rest is absolutely crystal clear now.
 
 
 
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