show that f(x)=sin 3x/tan 2x if x<0 ,3/2 if x=0 and log(1+3x)/e has power 2x minus1 if x>0 ; is continuous at x=o.

F(0) = 3/2 (given) --(1)

Limx->0⁺ f(x) =Limx->0⁺ Log(1+3x)/e^2x-1 = Limx->0⁺ log(1+3x)*3x /(3x) /(e^2x-1)*2x/(2x) =

Limx->0⁺ 3x/2x * Lim 3x->0(Log(1+3x)/3x ) / ( lim2x->0 (e^2x-1)/2x ) = 3/2*(1)/(1)= 3/2  --(2)

(as lim x->0 log (1+x)/x = 1 and lim x->0 (e^x-1)/x =1) 

and Lim x->0^- F(x) = lim x->0 sin 3x/tan2x = lim x->0 3x(sin 3x/3x ) / 2x(tan2x /2x) =

lim x->0 (3x/2x)lim3 x->0(sin 3x/3x ) / lim 2x->02x(tan2x /2x) = 3/2 *(1)/(1)  =3/2 ...(3)

from (10, (2) and (3) we have

Lim x->0⁺ f(x) = Lim x->0^- f(x) = 3/2 => lim x_0 f(x) = 3/2 which is same as f(0)

so f(x) is continuous at x= 0

(as lim x->0 sinx /x = lim x->0 tanx/x=1)