in other words, a a has the same meaning as Ax 0 Flip the script: Ax = xa is the same as x = a + Ax replacing x with a + Ax, our point B is instead: (a + Ax, f(a + Ax)) and Ay = f(x) - f(a) becomes Ay = f(a + Ax) f(a). also, instead of xa, we have Ax 0 representing the same idea this means lim Ay will instead be written as: lim Ay' 0 Ax This adjustment will make our calculations easier in some cases. Ax is cumbersome to write, so we frequently use h instead of Ax. so lim 0 f(a+Ax) f(a)` Ax -f(a)) is written more simply: lim h0 f(a+h) f(a)` h -f(a)) the benefit of this approach is that our denominator is now a monomial, which tends to be easier to reduce. The instantaneous slope of f(x) Start with two points A: (0,) and B: (0+ h, ?) What is the y-coordinate of B? Compute Ax and Ay Compute and Reduce Ax/Ay Finding the limit when f(x) is rational 1.0 (0,(1/2)) f(x) = 2 z+4 compute the instantaneous slope at: (0, 1)