For $0 < \mu \leqslant 1$, we consider $F_\mu$ defined by:
$$\forall y \in ]0, +\infty[, \quad F_\mu(y) = \frac{a}{\mu} y\left(1 - \left(\frac{y}{\theta}\right)^\mu\right)$$
and $F_0$ defined by $F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$, with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$.
(a) Show that $F_\mu$ converges pointwise to $F_0$ as $\mu$ tends to 0.
(b) Show that $\phi_\mu$ converges pointwise to $\phi_0$ as $\mu$ tends to 0.