Question #9a058

1 Answer
Sep 19, 2015

e

Explanation:

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)