Question #9a058

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Answer 1 · 2015-09-19T11:50:57

#e#

#lim_(x->0)((3+x)/(3-2x))^(1/x)=lim_(x->0)((3-2x+2x+x)/(3-2x))^(1/x)=#

#=lim_(x->0)(1+(3x)/(3-2x))^(1/x)=A#

Let #(3x)/(3-2x)=1/t#, then:

#3tx=3-2x => x(3t+2)=3 => 1/x=(3t+2)/3=t+2/3#

It's obvious that when #x->0# then #t->oo#.

#A=lim_(t->oo)(1+1/t)^(t+2/3)=#

#=lim_(t->oo)(1+1/t)^(2/3) * lim_(t->oo)(1+1/t)^t=#

#=1 * e=e#

Note:
#t->oo => 1/t->0#

#lim_(t->oo)(1+1/t)^t =e#

Note 2:

#lim_(x->0)((3-2x+2x+x)/(3-2x))^(1/x)=#

#lim_(x->0)((3-2x)/(3-2x)+(2x+x)/(3-2x))^(1/x)=#

#lim_(x->0)(1+(3x)/(3-2x))^(1/x)#