How do changes in α affect β for a BJT?
Alternatively to @Chu's answer, use sensitivity analysis to determine \$\beta\$'s sensitivity to changes in \$\alpha\$.

$$
S_{\alpha}^{\beta} =\lim_{\Delta \alpha \to 0}\left \{ \frac{\Delta \beta / \beta}{\Delta \alpha / \alpha} \right \}=\frac{\alpha}{\beta} \frac{\partial \beta}{\partial \alpha} = \frac {1}{1-a}
$$

Now, as @Chu has already mentioned, since \$\alpha \approx 1\$ (via the problem statement),

$$
\frac {1}{1-\alpha} \biggm \rvert_{\alpha \approx 1} \approx \frac {\alpha}{1-\alpha} = \beta
$$

So,

$$
\frac {\Delta \beta}{\beta} 
= \frac {1}{1-\alpha} \: \frac {\Delta \alpha}{\alpha}
\approx \frac {\alpha}{1-\alpha} \: \frac {\Delta \alpha}{\alpha} 
= \beta \: \frac {\Delta \alpha}{\alpha}
$$

for \$\alpha \approx 1\$ as \$\Delta \alpha \to 0\$.