180 Chapter 4 • Limits of Functions
Example 4.1.2 Consider the limit statement lim(4x - 5) = 3.
X->2
(a) Find a value of o > 0 that will guarantee that whenever x is within a
distance o from 2 (but not equal to 2) 4x -5 is within a distance .0 1 from
3.(b) Find a value of o > 0 that will guarantee that whenever x is within
distance o from 2 (but not equal to 2) 4x - 5 will approximate the limit
accurately to 3 decimal places.( c) For arbitrary r:: > 0, find a value of o > 0 that will guarantee that when-
ever x is within a distance o from 2 (but not equal to 2) 4x - 5 is within
a distance r:: from 3.Solution: This example should remind you of Example 2.1.5. You are
advised to take a look at that example. We begin by letting f(x) = 4x - 5.
(a) We want a real number O > 0 such that 0 < Ix - 21 < O::::} lf(x) - 31 <
.01. Now,
lf(x) - 31 = l(4x - 5) - 31 = l4x - 81
= 4lx-21.Thus, our objective is to find a o > 0 such that
0 < Ix - 21 < o::::} 4lx - 21 < .01.
The latter inequality will be true if Ix - 21 < -~l = .0025. Thus, we take
0 = .0025.
(b) This time, we want a real number 0 > 0 such that 0 < Ix - 21 < O::::}
lf(x) - 31 < .0005. Thus, our objective is to find a o > 0 such that
0 < Ix - 21 < O ::::} 4lx - 21 < .0005.
The latter inequ ality will be true if Ix - 21 < .0~0
5
= .000125. Thus, wetake o = .000125.
(c) Let r:: > 0. This time, we want a real number o > 0 such that 0 <
Ix - 21 < o::::} lf(x) - 31 < r::. Thus, our objective is to find a o > 0 such that
0 < Ix - 21 < O::::} 4lx - 21 < r::.
The latter inequality will be true if Ix - 21 < ~-Thus, we take o = ~- D
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