I need a clarification on the utility of Jordan's lemma. I think I have understood the theorem and its implications. It basically implies that if you have a function like $g(z) =f(z)e^{iz}$ it suffices for $f(z)$ to tend to zero at infinity in order to have a negligible integral of $g(z)$ along a sufficiently big semicircle in the upper half plane.
However, It confuses me that Wikipedia provides, as an example, the evaluation of this integral: $\int_{-\infty}^\infty \frac{\cos x}{1+x^2}\,dx$
Then it uses the fact that $f(z)=\frac{e^{iz}}{1+z^2}$ satisfies Jordan's lemma. But it seems to me that this particular function satisfies the Estimation lemma in the upper part of the Gauss graph, a stronger condition, and Jordan's lemma is not needed