By Arthur Cayley

This quantity is made out of electronic photographs from the Cornell collage Library ancient arithmetic Monographs assortment.

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Lim g(x) is just g(x). hS0 lim hS0 g(x ϩ h) Ϫ g(x) f(x ϩ h) Ϫ f(x) ϭ fr(x)!!!!! So lim ϭ gr(x)! hS0 h h Substituting, we get the product rule f(x)gЈ(x) ϩ g(x) fЈ(x). RULE 8 The quotient rule. If y ϭ g(x)fr(x) Ϫ f(x)gr(x) f(x) , then yr ϭ . g(x) [g(x)]2 39 40 BOB MILLER’S HIGH SCHOOL CALCULUS FOR THE CLUELESS The quotient rule says the bottom times the derivative of the top minus the top times the derivative of the bottom, all divided by the bottom squared. EXAMPLE 10— Find yЈ if y ϭ (bottom)(top)′ ؊ (top)(bottom)′ bottom squared yr ϭ x2 .

X ϩ 6| р |x| ϩ |6| Ͻ 5 ϩ 6 ϭ 11 Finishing our problem, we have |x ϩ 6||x Ϫ 4| Ͻ 11 . ␦ ϭ ε. So ␦ ϭ minimum (1, εր11). EXAMPLE 20— Prove 2 2 lim x ϭ 5 xS5 |2(5 Ϫ x)| 10 Ϫ 2x |2||5 Ϫ x| 2 2 ϭ P x Ϫ 5 P ϭ P 5x P ϭ |5 x| |5||x| ϭ 2|x Ϫ 5| 5|x| Again take a preliminary ␦ ϭ 1. |x Ϫ 5| Ͻ 1. So 4 Ͻ x Ͻ 6. To make a fraction larger, make the top larger and the bottom smaller. 0 Ͻ |x Ϫ 5| Ͻ ␦. We substitute ␦ on the top. Since x Ͼ 4, we substitute 4 on the bottom. 15 16 BOB MILLER’S HIGH SCHOOL CALCULUS FOR THE CLUELESS 2|x Ϫ 5| 2 # ␦ ␦ ϭε Ͻ5 # 4ϭ 10 5|x| ␦ ϭ 10ε If ␦ ϭ minimum (1, 10 ε), then |2/x Ϫ 2/5| Ͻ ε.

Informally, lim f(x) ϭ L means the larger x gets, the xS` closer f(x) gets to L. FORMAL DEFINITION Given an ε Ͼ 0, there exists a large positive number N such that whenever |x| Ͼ N, then |f(x) Ϫ L| Ͻ ε. In other words, for every small number ε, we must be able to find a large number N. For numbers bigger than N, f(x) will be very close (ε close) to L. The picture would look like this: Show 2x ϩ 5 lim x ϩ 1 ϭ 2 xS` y=L+ε y=L y = L –ε N The Beginning—Limits EXAMPLE 21— Suppose we are given ε Ͼ 0.