Question

Tricky limit?

Answer 1

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

Answer 2

sorry, thats kind of hard to see.

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

right.