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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. 1