202 Chapter 4 11 Limits of Functions
(b) lim [f(x)g(x)] exists, but lim f(x) and lim g(x) do not.
X-+Xo X-l-XO X-+Xo(c) lim [f((x))] exists, but lim f(x) and lim g(x) do not.
X->Xo g x X->Xo X->Xo
In each case, explain your answers.- Use the sequential criterion to prove Theorem 4.2.11 (a).
- Use the sequential criterion to prove Theorem 4. 2 .11 (b).
- Use the sequential criterion to prove Theorem 4.2.11 (e).
- Use the sequential criterion to prove Theorem 4.2.11 (f).
14. Prove Theorem 4.2.18.15. Suppose that f and g are defined in some N6(xo). Use Theorem 4.2.11 to
prove that if lim f((x)) exists and lim g(x) = 0, then lim f(x) = 0.
X-+Xo g x X-+Xo X-+Xo
16. Prove Theorem 4.2.20 (b).- Prove Theorem 4.2.22 (b).
- Consider. the funct10n. f(x) = { x. if x is.. rational. }. Prove that hm. J(x) =
-X if X is irrat10nal x->O
- Prove that if lim f(x) = 0, and g(x) is defined and bounded on a deleted
x-+xo
nbd. of x 0 , then lim [f(x)g(x)] = 0. (Compare with Exercise 2.2.6.)
x-+xo - Neighborhood Inequality Property of Limits, I: Prove the follow-
ing extension of Theorem 4.2.9. If lim f(x) > M then J(x) > M for all
X-+Xo
x in some deleted neighborhood of x 0. That is, if lim f(x) > M then
X-+XQ
::J deleted neighborhood N~(xo) 3 'ix E D(J) n N~(xo), f(x) > M. State
and prove a similar result that holds if lim f(x) = L < M.
X-+Xo - Neighborhood Inequality Property of Limits, II: P rove that if
lim f(x) < lim g(x), then f(x) < g(x) for all x in some deleted neigh-
x-+xo x-+xo
borhood of xo. - Cauchy Criterion for Limits of Functions: Suppose x 0 is a cluster
point of D(J). Prove that lim f(x) exists {::}-'le > 0, ::Jo > 0 3 'ix, y E
x-+xo
D(J), x, y E N6(xo) * lf(x) - f(y)I < c. - Suppose f : JR --> JR is periodic with period p > 0. That is, 'ix E JR,
f(x +p) = f(x). Prove that '<lxo E JR, lim f(x) = L {::}- lim f(x) = L.
X->Xo X->Xo+P