complex analysis - Find $intlimits_0^1 limlimits_{nrightarrowinfty}(nz^{n-1})mathrm{d}z$

The problem is to find: $$\int\limits_0^1 \lim\limits_{n\rightarrow\infty}(nz^{n-1})\mathrm{d}z$$

I started by finding $\lim\limits_{n\rightarrow\infty}(nz^{n-1})$. Naturally it converges to zero on $[0,1)$. However at $z = 1$, the limit diverges. So in essence on the contour of integration we have a zero function except for the endpoint, which is $\infty$. My question now is how do I integrate this?

I thought I could break it up into two integrals: $$\lim\limits_{\beta\rightarrow1^-}\left(\int\limits_0^\beta 0\mathrm{ d}z +\int\limits_\beta^1\lim\limits_{n\rightarrow\infty}(nz^{n-1})\mathrm{d}z\right)$$

My question now is does the integral of an infinite point equal zero? In the limit we get, essentially, $\int\limits_\beta^1\infty\mathrm{ d}z$. Usually a point integral is zero, but does that hold when the point itself is divergent to infinity?

I know the answer is supposed to be $0$, and that makes sense. I'm having trouble getting there rigorously, though.

Any help is greatly appreciated.

PS: This is for complex variables, but I realize the contour of integration is on the real line. I assume that would be the best way to do it. However, if you know of some fancy way to do it that strays from the real line feel free to do so, as this is complex variables after all.