Question

Limit problem ?

Hello,
I'm in trouble to calculate the limit as n->infinity of `ln(1+(3x)^n)/(n+5x)`
thank you

Answer 1

`lim_(n->oo)ln(1+(3x)^n)/(n+5x) = {(ln(3x), 3x ge 1),(0, 3x < 1):}`

Explanation:

`ln(1+(3x)^n)/(n+5x) = ln((3x)^n((3x)^-n+1))/(n+5x) =`

`=n/(n+5x) ln(3x) +ln((3x)^-n+1)/(n+5x)`

now for `3x ge 1 rArr lim_(n->oo)ln((3x)^-n+1)/(n+5x)=0`

and for `3x < 1 rArr lim_(n->oo)ln((3x)^-n+1)/(n+5x)=-ln(3x)`

hence

`lim_(n->oo)ln(1+(3x)^n)/(n+5x) = {(ln(3x), 3x ge 1),(0, 3x < 1):}`