I am looking at logarithms and derivatives. In my books, Bostock and Chandler, it saying:
$\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0}(\frac{1}{\frac{\delta x}{\delta y}})$
It then saying, "Now $\delta y\to 0$ as $\delta x \to 0$". I don't understand this part. How is $\delta y$ going to $0\,$?
We know that $\delta y \to 0$ because the derivative $\frac {dy}{dx}$ exists at the point in question.
If $\delta y$ does not go to $0$, then there is some $\epsilon > 0$ such that for every $\delta > 0$, there is a $\delta x$ with $|\delta x| < \delta$ but with $|\delta y| \ge \epsilon$, and therefore $\left|\frac{\delta y}{\delta x}\right| > \frac{\epsilon}{\delta}$. Since $\delta$ can be chosen arbitrarily small, $\left|\frac{\delta y}{\delta x}\right|$ is unbounded where $\delta x$ is near to $0$, and so the derivative does not exist.