Can a limit of form$\ \frac{0}{0}$ be rational if the numerator is the difference of transcendental functions, and the denominator a polynomial one? Let$\ f_1(x)$ and$\ f_2(x)$ be transcendental functions such that$\ \lim_{x\to 0} f_1(x)-f_2(x)=0$, and$\ f_3(x) $ polynomial, such that$\ f_3(0)=0$. Can$\ \lim_{x\to 0} \frac{f_1(x)-f_2(x)}{f_3(x)}$ be rational?

$$
\frac{\sin x -(e^x-1)}{x^2} \to - \frac12
$$
