Question

lim x->0 of (1+x)^(1/x)

Answer 1

the answer is e, but how do I get there?

Answer 2

one of two ways
the first way, which is probably in the book is to "take the log"
you get
\[\ln(1+x)^{\frac{1}{x}}=\frac{1}{x}\ln(1+x)\] take the limit of that

Answer 3

yeah, this limit is 1

Answer 4

you get \(1\) and so the answer to the original problem is \(e^1\)
that is one way

Answer 5

the other way is to recognize one definition of \(e\) as
\[\lim_{n\to \infty}(1+\frac{1}{n})^n\] or equivalently \[\lim_{x\to 0}(1+x)^{\frac{1}{x}}\]

Answer 6

third way to to write what
\[(1+x)^{\frac{1}{x}}\] really means, namely
\[\large e^{\frac{1}{x}\ln(1+x)}\] and take the limit there
it is the same work as the first way

Answer 7

ok, with the first case, my textbook says ln y = ln f(x), then find the lim is 1, then..
lim of ln y = 1
I dont get the next step
e^1, what happend to the lim of ln y ?

Answer 8

you took the log as the first step

Answer 9

get the limit is one
but that is the limit of the log of the function , not the limit of the original function
you have to go back to the original function
\[\lim_{x\to a}\ln(f(x))=L\implies \lim_{x\to a}f(x)=e^L\]

Answer 10

alright, thank you very much!