I'd like to compute the limit of
$$\mathop {\lim }\limits_{x \to +\infty } \frac{{\Gamma \left( {\frac{1}{2} - \frac{x}{2}} \right)\Gamma \left( {1 + \frac{x}{2}} \right)}}{{\Gamma \left( {\frac{1}{2} + \frac{x}{2}} \right)\Gamma \left( {\frac{x}{2}} \right)}}$$
Because of the physical problem involved, I can assume that x is even, i.e. x=2n (n positive non null integer) but... does it make sense to "approach infinity on even numbers"? Or this apparently oscillating function between +inf and –inf can somehow balance to zero? I’m little bit confused... Thanks in advance for any you can provide. Regards