Limit of an expression in a sequence with small ratio

Let $\{a_n\}_{n\ge1}$ be a sequence of positive real numbers such that $\lim_{n\to\infty}\dfrac{a_{n+1}}{a_n}=\dfrac12$. Then $\lim_{n\to\infty}\dfrac{e^{a_n^2}+a_n}4$ is equal to