4.2 Algebra of Limits of Functions 201
EXERCISE SET 4.2
- Prove Theorem 4.2.1.
- Prove Theorem 4.2.5.
- Prove Lemma 4.2.10.
- Prove Case 2 of Theorem 4.2.11 (a).
- Prove Theorem 4.2.11 (c).
- Prove Theorem 4.2.11 (f).
- Suppose lim f(x) = 2 and lim g(x) = 3. Find each of the following,
X--+Xo X--+Xo
and justify your answers:
(a) lim [4f(x) - g(x)]
X--+Xo (b) X-+Xo lim J f (x)g(x)
(c) lim Jf(x) - 3g(x)J
X--+Xo
( d) lim f (x)
x->xo f(x) + 2g(x)
- Use the "algebra of limits of functions" (Theorem 4.2.11 and 4.2.18 where
necessary) to find each of the following limits. Justify each step.
(a) lim (5x^2 + 8x + 7)
x->- 1
(b) lim(4x^6 + 3x^4 - 8)
X->0
(c) lim 3~2 - 91 (d) lim x2 - 3x + 2
X->3 X + X->2 X^2 + X - 2
( e) lim 3x - 9 ( f) lim x2 - 3x + 2
x-.3 x^2 - 9 x-.2 x^2 - x - 2
(g) lim f(x + h~ - f(x), where f(x) = 3x + 10
h->O
f(x + h) - f(x)
(h) ~~ h , where f(x) = x^2 - 5x
(i) lim V
5
x +
4
X->l X + 3
(k) lim _y'X_x_-_fa_a (a> 0)
X->a X - a
3 3
(m) lim x - a
X->a X - a
xn - l
(o) lim -- (n EN)
X->l X - 1
(j) lim (x + 1)2 - 1
X->0 X
(1) lim v'f+h -^1
h-.O h
x4 - a4
(n) lim--
x-.a x - a
xm - l
(p) lim -- (m,n EN)
x-.1 xn - 1
- In each of the following, give an example of functions f and g and a
cluster point xo of V(f) n V(g) satisfying the stated property:
(a) lim [f(x) + g(x)] exists, but lim f(x) and lim g(x) do not.
X-+Xo X--+Xo X--+Xo
