Use ML lemma to show $\lim_{R \to \infty} \int_{C_R} \frac{z^2}{z^4+1} dz =0$

Question

I am meant to use the ML lemma to deduce that $\lim_{R \to \infty} \int_{C_R} \frac{z^2}{z^4+1} dz =0$ where $C_R$ is the semi circle in the upper half of the complex plane oriented from $z=R$ to $z=-R$, centered a t the origin.

So I am trying to use $\vert \int_{C_R} \frac{z^2}{z^4+1}\vert dz \le \max(\vert f(z) \vert)*l$, where l is the arc length of $C_R$.

I find $\max(\vert f(z) \vert)=1/2$ and the arc length of the semicircle is $\pi R$. But this does not converge to zero as $R$ goes to infinity. I feel like I must be doing something obviously wrong but i cant seem to work it out.

Answer 1

Assume $R > 1$. Then $|z^2| = R^2$ and $|z^4 + 1| \ge R^4 - 1$, resulting in $$ \left| \frac{z^2}{z^4+1} \right| \le \frac{R^2}{R^4-1}. $$ Now you have $$ \frac{R^2}{R^4-1} \cdot \pi R \xrightarrow{R \to \infty} 0, $$ as needed.