1. limits

Hi!!

I'm very confused about limits.. Can someone please correct my working out? Because I'm very sure that I'm wrong, but I don't know what to do. Thank you

2. Originally Posted by gconfused
Hi!!

I'm very confused about limits.. Can someone please correct my working out? Because I'm very sure that I'm wrong, but I don't know what to do. Thank you
the second part just becomes 1 as n->infinity.

the first part:

$\lim_{n->inf}(\frac{1}{1+y/n})^{n-1}$

$\huge e^{\lim_{n->inf}(\frac{1}{1+y/n}-1){n-1}}$

$\huge e^{\lim_{n->inf}(\frac{n}{(n+y)}-1)({n-1})}$

$\huge e^{\lim_{n->inf}(\frac{n-1}{n+y})({-y})}$

$\huge e^{-y}$

3. Only the last step is wrong

lim n->oo 1/( 1+ Y/n)^n * [1 + Y/n]^-1
= lim n->oo 1/( 1+ Y/n)^n * lim n->oo [1 + Y/n]^-1
= 1/ e^Y * 1
= exp(-Y)

4. Thank you guys

5. It's easier to see the result if you make the substitution $\displaystyle k=\frac{y}{n}$ where $\displaystyle k \to 0$ when $\displaystyle n \to \infty$

$\displaystyle \lim_{k \to 0}(1+k)^{-y/k}=\lim_{k \to 0}e^{ln{(1+k)^{-y/k}}}=\lim_{k \to 0}e^{-y \cdot \frac{ln(1+k)}{k}}=e^{-y}$