Ok so my lecturer gave us this powerful lemma for doing contour integrals over a semi-circle. The lemma is:
Let $C_{R}$ be the contour defined as $\{z \in \mathbb{C} \mid z = R e^{i \theta } , \theta \in [0, \pi] \}$.
If $|f(z)| \le \frac{k}{ |z| ^2} $ for some $ k \in \mathbb{R} $ , $\Im(z) > 0$ and $|z|$ large enough, then:
$\lim_{R \to 0} \int_{C_{R}} f(z) dz = 0$
So in a lecture we did this problem:
$ \int_0^{\infty} \dfrac{dx}{ \sqrt{x} (1 + x) }$
He then defined $\Gamma _{R} = C_R \cup [\epsilon,R] \cup (-C_{\epsilon}) \cup [R,\epsilon]$
Where $C_R = \{z \in \mathbb{C} \mid |z| = R\}$, $C_{\epsilon} = \{z \in \mathbb{C} \mid |z| = \epsilon \}$
He somehow used the lemma to show that:
$\lim_{R \to 0} \int_{C_{R}} f(z) dz = 0$
But he can't do that can he?