Limit sup of complex function

Question

I wold like to determine this kind of limit sup: $$ \lim \sup_{\mid z \mid \to \infty } \left \{ \frac{\mid b(t)\mid}{\mid z - a(t)\mid} \right \} $$ Where $z \in \mathbb C$ such that $\mathrm{Re}z \geq 0$, $a$ and $b$ are two continuous real functions in some bounded interval $I$. What I have do: \begin{align*} \lim \sup_{\mid z \mid \to \infty } \left \{ \frac{\mid b(t)\mid}{\mid z - a(t)\mid} \right \}&= \lim_{\mid z \mid \to \infty } \left[\sup \left \{ \frac{\mid b(t)\mid}{\mid z' - a(t)\mid}, \mid z' \mid > \mid z \mid \right \} \right] \\ &\leq \lim_{\mid z \mid \to \infty } \left[\sup \left \{ \frac{\mid \max ( b(t))\mid}{\mid z'\mid - \mid \max ( a(t))\mid}, \mid z' \mid > \mid z \mid \right \} \right] \\ &= \lim_{\mid z \mid \to \infty } \left( \frac{\mid \max ( b(t))\mid}{\mid z\mid - \mid \max ( a(t))\mid} \right) \\ &=0 \end{align*} I have some confusion about this calcul or is there another method? Any help is welcome.

Answer 1

No. It's just fine. You should have used the word “computation” instead of “calcul”. The rest is just as it should be.