a limit

**dannee** · February 12th 2010, 02:57 PM

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 !

**skeeter** · February 12th 2010, 03:00 PM

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 ...

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

... and use L'Hopital's rule

**dannee** · February 12th 2010, 03:08 PM

great , thanks !

**xalk** · February 12th 2010, 11:18 PM

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 !

$lim_{x\to\infty}xln(1+\frac{1}{x}) = lim_{x\to\infty}ln(1+\frac{1}{x})^x$ = $ln(lim_{x\to\infty}(1+\frac{1}{x})^x) = lne=1$

**Noxide** · February 13th 2010, 12:18 PM

Originally Posted by xalk

$lim_{x\to\infty}xln(1+\frac{1}{x}) = lim_{x\to\infty}ln(1+\frac{1}{x})^x$ = $ln(lim_{x\to\infty}(1+\frac{1}{x})^x) = lne=1$

i don't understand what you did here at all

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

**skeeter** · February 13th 2010, 12:30 PM

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 ...

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

also ...

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

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