What is the difference between $\Delta f/\Delta x$ and $\Delta f(x)/\Delta x$? Are they the same?
I've been watching a lecture and the professor seems to describe the slope $$m = \lim_{x\to0} \Delta f/\Delta x$$
Is this the same as $\Delta y/\Delta x$?
Think geometrically: if $y$ varies independently of $x$, this means that we have a vertical line, and we can't talk about slope! The same idea applies in the limit case.
Thus, short answer: They are the same. The advantage of $\Delta f(x)$ over $\Delta f$ is that it explicits $f$ depending on $x$. For the limit $$m = \lim_{\Delta x \to 0} \frac{\Delta f}{\Delta x}$$ to exist we need to have $\Delta f$ as a function of $\Delta x$ such that $\Delta f$ also goes to zero as $\Delta x$ reaches zero.
In other words: If we want the limit to represent a slope it has to exist, and we need the numerator functionally related (e.g. the numerator's value depends on the denominator's) to the denominator.