a limit

• Feb 12th 2010, 02:57 PM
dannee
a limit
hello everyone, i need help to find the limit of this function: x*ln(1+(1/x)) when x going to infinity. can anyone explain please how should i deal with this one? thanks !
• Feb 12th 2010, 03:00 PM
skeeter
Quote:

Originally Posted by dannee
hello everyone, i need help to find the limit of this function: x*ln(1+(1/x)) when x going to infinity. can anyone explain please how should i deal with this one? thanks !

rewrite as ...

$\displaystyle \frac{\ln\left(1+\frac{1}{x}\right)}{\frac{1}{x}}$

... and use L'Hopital's rule
• Feb 12th 2010, 03:08 PM
dannee
great , thanks !
• Feb 12th 2010, 11:18 PM
xalk
Quote:

Originally Posted by dannee
hello everyone, i need help to find the limit of this function: x*ln(1+(1/x)) when x going to infinity. can anyone explain please how should i deal with this one? thanks !

$\displaystyle lim_{x\to\infty}xln(1+\frac{1}{x}) = lim_{x\to\infty}ln(1+\frac{1}{x})^x$ =$\displaystyle ln(lim_{x\to\infty}(1+\frac{1}{x})^x) = lne=1$
• Feb 13th 2010, 12:18 PM
Noxide
Quote:

Originally Posted by xalk
$\displaystyle lim_{x\to\infty}xln(1+\frac{1}{x}) = lim_{x\to\infty}ln(1+\frac{1}{x})^x$ =$\displaystyle ln(lim_{x\to\infty}(1+\frac{1}{x})^x) = lne=1$

i don't understand what you did here at all

lim as x approaches 0 of ln(1+x)^(1/x) = lne = 1
• Feb 13th 2010, 12:30 PM
skeeter
Quote:

Originally Posted by Noxide
i don't understand what you did here at all

lim as x approaches 0 of ln(1+x)^(1/x) = lne = 1

it's a definition ...

$\displaystyle \lim_{x \to 0} (1+x)^{\frac{1}{x}} = e$

also ...

$\displaystyle \lim_{n \to \infty} \left(1+ \frac{1}{n}\right)^n = e$