Question

in your own words, describe how limits can be used to find the instantaneous rate of change of a function.

Answer 1

Well, supposed to be your words, not mine.
If a variable y depends on x as a continuous smooth function y = f(x) and we want to find the rate of change of y as x changes then
on a graph, that is the slope of a tangent to the function, change in y / change in x at the point (x,y).
so how to find that slope is the question.
well
slope = [y2 - y1 ]/(x2-x1)
That slope depends on how y changes with x and is only reliably tangent to the curve as x2-x1 --->0 (sketch graph)
if we call x2-x1 = dx
now at the point (x,y)
x2 = x1 + dx
then y2 = y1 + slope*(dx)+
then [y2-y1/[x2-x1] =slope*dx/dx = slope.
Write it your way :)