Does $\int_{-1}^1\frac{1}{x}dx$ Converge or diverge?
Is the following a valid proof of convergence?:
\begin{align} &= \lim_{a\to0+}\left(\int_{-1}^{-a}\frac{1}{x}dx + \int_{a}^1\frac{1}{x}dx\right)\\ &= \lim_{a\to0+}\left([\ln|x|]_{-1}^{-a} + [\ln|x|]_{a}^{1}\right)\\ &= \lim_{a\to0+}(\ln|-a| - \ln|a| + \ln|1|-\ln|-1|)\\ &= \lim_{a\to0+}\left(\ln\left|\frac{-a}{a}\right|\right)\\ &= \ln|-1| = \ln{1} = 0 \end{align}
Thanks.