# 3.1 The Limit of the Difference Quotient # Art Belmonte, Summer 1996 # These are the shaded gray examples which appear in Section 3.1. > restart; f:=x->x^3 - 8*x; > Limit((f(2+h) - f(2)) / h, h=0); value("); 3 f := x -> x - 8 x 3 (2 + h) - 8 - 8 h lim ------------------ h -> 0 h 4 > y:=4*(x - 2) - 8; y := 4 x - 16 > plot({y, f(x)}, x=0..4); > plot({y, f(x)}, x=1.8..2.2); > Limit((f(x+h) - f(x)) / h, h=0); > Df:=value("); subs(x=2, Df); 3 3 (x + h) - 8 h - x lim ------------------- h -> 0 h 2 Df := -8 + 3 x 4 > diffq:=(f(x+h) - f(x)) / h; 3 3 (x + h) - 8 h - x diffq := ------------------- h > simplify(diffq); 2 2 3 x + 3 x h + h - 8 >