[!]. This exercise is used in Sections 5.4 and 6.1. Define L : (0,00) --+ R by L(x) = t dt. J1 t

(a) Prove that L is differentiable and strictly increasing on (0,00), with L'(x) =

l/x and L(l) = o.

(b) Prove that L(x) --+ 00 as x --+ 00 and L(x) --+ -00 as x --+ 0+. (You may wish to prove that for all n EN.)

(c) Using the fact that (xq ), = qxq- 1 for x > 0 and q E Q (see Exercise 8, p.

94), prove that L(xq ) = qL(x) for all q E Q and x> o.

(d) Prove that L(xy) = L(x) + L(y) for all x, y E (0,00).

(e) Let e = limn -+oo (l + l/n)n. Use I'Hopital's Rule to show that L

(e) = l.

(L(x) is the natural logarithm function log x.)

00. This exercise was used in Section 4.3. Let E = L -1, where L is defined in Exercise 4.