Answer by Andreas H. for What does it mean to have a pole or zero at infinity?

Suppose we have a system with system response of \$H(s)\$ (i.e. the Laplace transform of the impulse response).

Then \$H(s)\$ has a zero/pole at infinity if the function $$H(1/s)$$ has a zero/pole at \$s = 0\$. This is a definition, so there is no derivation.

If there is a pole at infinity, this means that the frequency response \$H(i\omega)\$ is going to infinity for \$\omega \to \infty\$, which may make the system unstable. An example is the e.g. the differentiator, which has transfer function \$s\$ and hence a pole at infinity.