Define $L:(0,\infty)\to\mathbb{R}$ by
$L(x)=\int_{1}^{x}\frac{dt}{t}.$
Show that $\lim_{x\to0^{+}}L(x)=-\infty$ and $\lim_{x\to\infty}L(x)=\infty$.
This is what I have done: We have that
$\lim_{x\to0^{+}}L'(x)=\lim_{x\to0^{+}}\frac{1}{x}=\infty.$
So, it must be the case that $\lim_{x\to0^{+}}L(x)=\pm\infty$. Moreover, we have that
$L(x)=\int_{1}^{x}\frac{dt}{t}=-\int_{x}^{1}\frac{dt}{t}.$
Hence, this implies that $\lim_{x\to0^{+}}L(x)=-\infty$.
For the other case, we have that
$\lim_{x\to\infty}\int_{1}^{x}\frac{dt}{t}>1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$
But since this sum does not converge, it must be the case that $\lim_{x\to\infty}L(x)=\infty$.
What do you guys think?