Why does the integral of 1/x from negative infinity to infinity diverge?

Question

I'm confused as to why $$\int_{-\infty}^\infty\frac{1}{x}dx$$ diverges. If $\frac{1}{x}$ is an odd function shouldn't the area to the left of origin be the opposite of the area to the right of origin, resulting in a net area of 0?

Answer 1

As an improper integral, $\int_{-\infty}^{\infty} \frac{1}{x} dx = \lim_{N \to \infty} \int_{-N}^{ - \frac{-1}{N}}\frac{1}{x} dx + \lim_{M \to \infty} \int_{\frac{1}{M}}^{M}\frac{1}{x} dx$