Define $L(x)=\int_{1}^x {1\over t} dt $
NOTE: I realize that $L(x)$ is the definition of $ln(x)$, but we aren't allowed to use that. Our professor is walking us through the definition of $ln(x)$ and $e^x$.
Part A: Show that $L({1\over x}) = -L(x)$
I've tried several different substitutions for this and even direct proof, but I'm completely stuck after working/thinking about this for the past several hours. Help is greatly appreciated!
Part B: Using Cauchy Criterion showing that the sequence $s_n = 1 + {1 \over 2} + {1 \over 3} + ... + {1 \over n}$ is divergent if $m>n$, show that $L(x)$ tendsto $\infty$ as $x \to \infty$.
Basically, I'm trying to show that if the limit of the sequence converges to some L, then the function of that sequence also converges to the same L. I suspect that I need to show that $l(x)=s_n={1 \over x}$, but I don't know how to formally state this idea.