Extra Credit 4 (5 pts):
1. Recall that the definition of the derivative of f (x) at the point (x0 , f (x0 )) is
0
f (x0 ) = lim
x→x0
f (x) − f (x0 )
f (x0 + h) − f (x0 )
= lim
.
h→0
x − x0
h
We saw a picture in class that showed that the two limits above are in fact calculating
the same thing. To get the extra credit points I want you to prove this algebraically. So
write down a string of equalities that show
lim
x→x0
f (x) − f (x0 )
f (x0 + h) − f (x0 )
= lim
.
h→0
x − x0
h
(Hint) All this takes is the right substitution. Start with the left limit then relabel the
point (x, f (x)) as something else. In other words, if you plug the right thing in for x in
the left limit, it should turn into the right limit.
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