Hello crymorenoobs

Welcome to Math Help Forum! Originally Posted by

**crymorenoobs** Hi, I'm trying to work out (limit as n=>infinity) (n * (sqrt(1 + 1/n) - 1)).

I know that by letting n = 1/t and then using l'hopitals this equals 1/2.

However, I need to use only basic limit rules (add/sub/mult/div and composition).

I feel like I am really close to the answer because I have been able to rewrite this as (n * (1 + 1/n)^(1/2)) - n.

I noted that:

ln(n * (1 + 1/n)^(1/2))) - ln(n)

= ln((1 + 1/n)^(1/2))

= 1/2(ln(1+1/n))

and since lim(1+1/n) = e as n=>inf

= 1/2(ln(e))

= 1/2

I cannot figure out any way to actually bring the ln into the equation like that though...any thoughts? Suggestions would be much appreciated.

What about the Binomial Expansion?$\displaystyle \left(1+\frac1n\right)^{\frac12}$$\displaystyle =1+\left(\frac12\right)\left(\frac1n\right)+\frac{ 1}{2!}\left(\frac12\right)\left(-\frac12\right)\left(\frac1n\right)^2+\frac{1}{3!}\ left(\frac12\right)\left(-\frac12\right)\left(-\frac32\right)\left(\frac1n\right)^3+...$

Subtract $\displaystyle 1$, multiply by $\displaystyle n$, and you're there.

Grandad