Limit problem ?

Question

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

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Answer 1 · 2018-04-17T08:24:53

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

#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):}#