Don't cross your limits!

Since the limit is an indeterminate form $\frac{\infty }{\infty },$ it can be applied the L'Hôpital rule. By evaluating the derivative on the numerator and the denominator, we obtain:

$\large \displaystyle\lim_{x \rightarrow 0^{+}} \dfrac{ p + e^{\frac{1}{x}}}{ 1 + qe^{\frac{1}{x}} } = \dfrac{\frac{-1}{x^2} e^{\frac{1}{x}}}{q \frac{-1}{x^2} e^{\frac{1}{x}} } = \dfrac{1}{q} = 2,$ which then follows that $q = \dfrac{1}{2}$