4.2 Algebra of Limits of Functions 187- Prove each of the following limit statements using Definition 4.1.1:
(a) lim (2x + 13) = 5
X-->-4
(b) lim x^2 = 4
X-->-2
(c) x-->i lim(3x^2 + 2x - 3) = 2 (d) X-->-2 lim (3x^2 - 2x) = 16
(e) lim (2x^2 + 5x + 1) = 4
X-->- 3
(f) lim x^3 = -1
X--+-l
(g) lim ( x^3 + 5x) = 6
X-->i
x^2 -1
(h) lim --= -2
x-->-i X + 1
X^2 - 2x - 3
(j) lim = 4
X-->3 X - 3(i) lim^3 x2-^12 = -12
X-->-2 X + 2
3. Use the sequential criterion for limits of functions to prove each of the
limit statements given in Exercise 2.
4. In each of the following, a function f(x) and a number x 0 are given. Use
the sequential criterion to prove that lim f(x) does not exist.
(a) f(x) = kl ; xo = 0.
xX--+Xo(b) f(x) = { 5, ~f x < 3}; Xo = 3.
6, if x ::'.'. 3( ) c D iric .. hl e t' s Fu nc t⢠ion: f( x ) = { 1, if x is rational } ; x 0 E m JN..
0, if x is irrational- Prove that for constants a and b E JR, and 't/x 0 E JR, lim (ax+b) = ax 0 +b.
X--+Xo - Prove that lim f(x) = L {:} lim f(xo + h) = L.
X-->Xo h-->0
. { x if x is rational }.. - Prove that the function f(x) =.... has a hmit at xo
- x if x is irrational
if and only if xo = 0.
- Prove that lim !'4x^2 - 4x - x3 does not exist. [See 4.1.1, Note (2).]
X-->2 - Prove that lim !'x4 - 4x3 + 5x^2 - 2x does not exist. [See 4.1.1, Note (2).]
x-->i - Use Exercise 2.6.20 and the sequential criterion for limits of functions to
prove that lim sin (.!.) does not exist.
x-->O X
4.2 Algebra of Limits of Functions
The material of this section runs parallel with the algebra of limits of sequences,
as presented in Section 2.2. There are some differences, primarily due to the
fact that when given an E: > 0 we must find a 6 > 0 instead of an no EN, and