How to evaluate this rational limit Evaluate:
$$\lim_{x\to0^+}\frac{(-3\sqrt{x}+x^3+\sin(x^6))(\text{e}^{4\sqrt{x}}-1)}{4\ln (1+x)+x^4-x^6}$$

Exploiting the Taylor series of a function at the point $x=0$, we have 
$$ \frac{(-3\sqrt{x}+x^3+\sin(x^6))(\text{e}^{4\sqrt{x}}-1)}{4\ln(1+x)+x^4-x^6}$$
$$=\frac{(-3\sqrt{x}+x^3+(x^6-\frac{x^{12}}{3!}-\dots )))( (1+4\sqrt{x}+\dots)-1)}{4 (x+\frac{x^2}{2}\dots)+x^4-x^6}$$
$$ \sim \frac{(-3\sqrt{x})(4\sqrt{x})}{4x} = -3. $$
