What is the difference in
a.) f(a+Δx)-f(a)/Δx
and
b.) limit as Δx->0 f(a+Δx)-f(a)/Δx
I assume you're asking what is the difference between
$\displaystyle \frac{f(a + \Delta x) - f(a)}{\Delta x}$
and
$\displaystyle \lim_{\Delta x \to 0}\frac{f(a + \Delta x) - f(a)}{\Delta x}$.
The difference is that in the second you're asked to describe what happens as you make $\displaystyle \Delta x$ get close to $\displaystyle 0$.
Draw a picture showing the graph of a function. I presume you are allowed to use whatever function you want. (a, f(a)) is some point on that graph. $\displaystyle (a+ \Delta x, f(a+ \Delta x))$ is another point on that graph. $\displaystyle \frac{f(a+\Delta x)- f(a)}{\Delta x}$ is the slope of the line through those two points. Show that by drawing the line. Do that for several different values of $\displaystyle \Delta x$ so that you have several lines with one end at (a, f(a)). Do you see that the lines become closer and closer to the tangent line at (a, f(a)) as $\displaystyle \Delta x$ becomes smaller and smaller?
On a set of axes label two point (a, f(a)) and (a+delta x, f(a+delta x))
The first is just the gradient formula between those two points (rise over run) so it represents the gradient of the line going through those points.
Now think about what happens as you make delta x smaller and smaller. What happens to the line through those two points?