Math Help - a limit

1. 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 !

2. 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

3. great , thanks !

4. 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 !
$lim_{x\to\infty}xln(1+\frac{1}{x}) = lim_{x\to\infty}ln(1+\frac{1}{x})^x$ = $ln(lim_{x\to\infty}(1+\frac{1}{x})^x) = lne=1$

5. 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

6. 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 ...

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