4.4 *Infinity in Limits 223
Complete the proof of Example 4.4.22.
State and prove a sequential criterion for lim f ( x) = L (or ±oo)
x->+=
and a sequential criterion for lim f(x) = L (or ±oo).
X--+-00
Investigate each of the following:
(a) lim xsinx (b) lim .!_ sin x
X-+00 X--+00 X
. 1. 1
(c) hm - sm -
X->CXl X X
Suppose f is defined on a neighborhood of +oo and lim xf(x) =LE JR.
X->CXl
Prove that lim f(x) = 0.
X->CXl
Suppose f and g are defined on a neighborhood of +oo, g is positive on
this interval, and lim f((x)) = L E JR. Prove:
X->CXl g X
(a) if L > 0, then lim f(x) = +oo <=? lim g(x) = +oo;
X--+00 X--+00
(b) if L < 0, then lim f(x) = -oo <=? lim g(x) = +oo.
x--+oo x-+oo
Apply Theorem 4.4.24 to find each of the following:
(a) lim (5x^6 - 12x^5 + 2x^3 - 87) lim (5x^6 - 12 x^5 + 2x^3 - 87)
x--++oo x-+-oo
(b) x-++oo lim (13x^7 + 8x^4 - 7x^3 + 35) x--+-oo lim (13x^7 + 8x^4 - 7x^3 + 35)
(c) lim (9-x^2 +4x^3 -7x^11 ) lim (9-x^2 +4x^3 -7x^11 )
x--++oo x--+- oo- Apply Theorem 4.4.26 to find the horizontal asymptote(s) for the graph
of each of the following rational functions:
(a)f(x)= x+2 (b)f(x)=x2-3x+l
3x -1 x + 8
7x - 5 1-9x
(c) J(x) = 4x 2 + 3x - 7 (d) f(x) = x 2 + 4(e) f(x) = x3 - 5x (f) f(x) = 6x4 + 13 x2
4x^2 + 1 11-x^4
Prove that Theorem 4.4.19 remains true if Lis +oo or -oo.
Prove the following monotone convergence theorem: if f is monotone
increasing on a neighborhood of +oo, say (a, oo), then lim f(x) exists<=?
X->CXl
f is bounded above on (a,oo); and in this case, lim f(x) = supf(a,oo).
X->CXl
State similar results for monotone decreasing functions, and for x -> -oo.
Cauchy Criterion for Limits of Functions at Infinity: Suppose
D(f) is unbounded above. Prove that lim f(x) exists iff Ve: > 0, 3 N >
X->CXl
0 3 \fx, y E D(f), x, y > N => lf(x) - f(y)I < c:.
State a similar theorem for X-+-00 lim f ( x).