Divergence and Convergence of improper integrals of $1/x$ and $1/x^2$

Question

Prove that $\int_1^\infty dx/x $ diverges and $\int_1^\infty dx/x^{2} $ converges

I think that the former, $dx/x$ converges as plugging the bounds doesn't yield a non-existent result.

Answer 1

First integral: $$ \lim_{t\to\infty}\int_{1}^{t} \frac{dx}{x}=\lim_{t\to\infty}\ln(t)-\ln(1)=\infty. $$ Second: $$ \lim_{t\to\infty}\int_{1}^{t} \frac{dx}{x^2}=\lim_{t\to\infty}\frac{1}{-t}-\frac{1}{-1}=1. $$