当x趋向于正无穷时lim3xf(x)=lim[4f(x)+6],求当x趋向与正无穷时limxf(x)=?

Question

当x趋向于正无穷时lim3xf(x)=lim[4f(x)+6],求当x趋向与正无穷时limxf(x)=?

Answer 1

lim(x->+∞) 3xf(x) =lim(x->+∞) [4f(x)+6]
lim(x->+∞) { 3xf(x) - [4f(x)+6] } = 0
lim(x->+∞) (3x-4)*f(x)
=lim(x->+∞) [(3x-4)*f(x)-6 + 6]
=lim(x->+∞) [(3x-4)*f(x)-6 ] + lim(x->+∞) 6
=6
∴
lim(x->+∞) x*f(x)
=lim(x->+∞) {(3x-4)*f(x)} * {x/(3x-4)}
= 6*(1/3)
= 2