Show that `f(x)={(sin3x)/(tan2x),\ \ \ if\ x<0 3/2,\ \ \ if\ x=0(log(1+3x))/(e^(2x)-1),\ \ \ if\ x >0` is continuous at `x=0`

Explanation

(i) `f(0)=frac{3}{2}`


(ii) `lim _{x -> 0} f(x)`


Now, `lim _{x -> 0^{+}}(frac{log (1+3 x)}{e^{2 x}-1})`



`=lim _{x -> 0^{+}}(frac{frac{log (1+3 x)}{3 x}}{frac{e^{2 x}-1}{2 x}} times frac{3}{2})`




`=frac{3}{2}`


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