baldymcgee6
Tricky limit?



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baldymcgee6
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\[\lim_{x\to 0}\left(\frac{(1+x)^{\frac{1}{x}}}{e}\right)^{\frac{1}{x}}.\]

baldymcgee6
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sorry, thats kind of hard to see.

baldymcgee6
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\[\LARGE \lim_{x\to 0}\left(\frac{(1+x)^{\frac{1}{x}}}{e}\right)^{\frac{1}{x}}.\]

wio
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\[ \Large
e = \lim_{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^n
\]When you reparameterize: \(x=1/n\) \[\Large
e = \lim_{x \rightarrow 0}\left(1+x\right)^\frac{1}{x}
\]

wio
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So it's a matter of settling that outer 1/x

baldymcgee6
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I haven't learned 'reparameterize' yet, so I would assume I wouldn't have to use that... Supposed to use L'Hospital's rule

wio
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Okay, then you need to have an indeterminate form of \(\infty /\infty\) or \(0/0\)

baldymcgee6
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right.

mahmit2012
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dw:1353045336700:dw

mahmit2012
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dw:1353045499277:dw

baldymcgee6
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@mahmit2012 I am supposed to use L'Hospital's rule.

wio
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I can't read that bottom line.

baldymcgee6
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"and in this case it is 1/sqrt(e)

wio
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No, of the previous picture

baldymcgee6
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I don't know

wio
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Okay, let's start with bringing in the \(1/x\) to the numerator and denominator and figuring out if that is an indeterminate form.

mahmit2012
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dw:1353047217870:dw

mahmit2012
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dw:1353047297672:dw

baldymcgee6
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@waterineyes can you help me understand this maybe?