Take the improper integral
$\int^1 _{-\infty} \cos \pi x \; dx $
From which it is clear that:
$\lim_{b \to -\infty} \int^1 _{b} \cos \pi x \; dx = -\frac{1}{\pi}\sin b \pi$
The integral oscillates between $\frac{1}{\pi}\text{ and }-\frac{1}{\pi}$ as $b \to \infty$.
Now, my textbook, Calculus: One and Several Variables, by Salas et. al (10th ed.), says this integral "diverges"? Certainly, I agree it "does not converge", but I don't think the integral is diverging per se...
Is the diverge/converge distinction a mutually exclusive dichotomy? It seems like it should be possible to say an integral neither converges or diverges, in the situation that the function oscillates to infinity.