Limit and Derivative Problems with Infinity: Can Someone Explain?

[hytuoc]

Would some one please show me how to do these problems below.
Thanks so much
1) lim of n as n approaches to infinity 1/[1-(n/(n-1))]
2) lim of n as n approaches to infinity (n-1)/[n-(1/n)]
3) lim of n as n approaches to infinity n * sin (1/n)
4) lim of n as n approaches to infinity (1/n)-(1/(n+1))

****Would u please also explain this to me:
Y does lim of x as x approaches to 0+ {[1+(x/2)]^(2/x)}^(1/2) = e^(1/2)?
I don't understand y they r equal??

 

[hytuoc]

Would some one please show me how to do these problems below.
Thanks so much
1) lim of n as n approaches to infinity 1/[1-(n/(n-1))]
2) lim of n as n approaches to infinity (n-1)/[n-(1/n)]
3) lim of n as n approaches to infinity n * sin (1/n)
4) lim of n as n approaches to infinity (1/n)-(1/(n+1))

****Would u please also explain this to me:
Y does lim of x as x approaches to 0+ {[1+(x/2)]^(2/x)}^(1/2) = e^(1/2)?
I don't understand y they r equal??

 

[JonF]

1) lim of n as n approaches to infinity 1/[1-(n/(n-1))]

look at the n/n-1 part first.

Times both the top and the bottom by 1/n. That gives you 1/(1-1/n) The 1/n term goes to zero, so this part = 1

that leaves you with lim as x goes to 1 of 1/1-x, which is means the denominator goes to zero, so the equation goes to infinity.

Why don't you post what part you are stuck on with the rest?