Let the function $f(z)$ is continuous on $|z-z_0|>r_0$ and $M(r)$ is the maximum of $|f(z)|$ on the circle $|z-z_0|=r>r_0$. Suppose that $rM(r) \rightarrow 0$.
Prove that $\int_{C_r} {f(z)} \,\mathrm{d}z \rightarrow 0$ if $r \rightarrow \infty$. Here $C_r$ is the circle $|z-z_0|=r$.
The link to the similar exercise:"Prove that $\int_{C_r} {f(z)}\,\mathrm dz→0$ if $r \rightarrow 0$, where $C_r$ is the circle $|z-z_0|=r<R$"
But here the conditions are changed: $r \rightarrow \infty$ and the cirle $|z-z_0|=r>r_0$.
If the circle has the length $L$ then $\big|\int_{C_r}f(z)\,\mathrm dz\big| \leq M(r)*L = 2\pi*r*M(r)*L$. How to show then that $\int_{C_r} {f(z)}\,\mathrm dz→0$ if $ r \rightarrow \infty$?
Since$$\left|\int_{C_r}f(z)\,\mathrm dz\right|\leqslant2\pi rM(r),$$you have$$0\leqslant\lim_{r\to\infty}\left|\int_{C_r}f(z)\,\mathrm dz\right|\leqslant2\pi\lim_{r\to\infty}rM(r)=0.$$