Limit problem ?

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

1 Answer
Apr 17, 2018

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):}