Take the example $f(x) = x^{2/5}$. If you choose some $x_0 \ne 0$, then $x_1$ is given by$$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = -\frac 32 x_0.$$
So, you can see that the magnitude of the sequence elements increases by $\frac 32$ each iteration. Hence, $|x_n| = |x_0| (3/2)^n$ and you can conclude that the sequence never converges to zero.