Calculating a limit using dominated convergence theorem

I am trying to show that for $n \in \mathbb{N}$ and $x \in [-a,0]$, $$\lim_{x\rightarrow 0^-} \int_{0}^{1} x z^n e^{-xz} dz = \int_{0}^{1} \lim_{x\rightarrow 0^-} x z^n e^{-xz} dz = 0,$$ using dominated convergence theorem.

My attempt is as follow:

Define $f_x(z) = xz^n e^{-xz}$, then $$\lim_{x\rightarrow 0^-} f_x(z) = 0 = f(z).$$

To have a dominating function, we have $$\sup_{z\in[0,1]}|f_x(z) - f(z)| = \sup_{z\in[0,1]}|xz^n e^{-xz}|.$$

But I'm having problems determining a dominating function.

I need a hint about where to go from here.

The hint is that your function is continuous.