Why is the integral from $0$ to $\infty$ of $(\phi(x)-\phi(-x))/x$ well defined if $\phi$ is a smooth function with compact support?

Question

I am trying to understand the proof that the principal value of 1/x is a distribution :

Ok so I understand that we decomposed $\phi$ into its even and odd part and the odd part disappeared so we are left with the even part. The part I don't understand exactly is why we are able to say that $\lim \epsilon \to 0^+$ of $\int_{\varepsilon}^\infty \frac{\varphi(x)-\varphi(-x)}{x}\textrm{d}x$ is in fact $\int_{0}^\infty \frac{\varphi(x)-\varphi(-x)}{x}\textrm{d}x$. How do we know the integral does not blow up to infinity as we take the limit $\epsilon \to 0$?