Page 74 - Calculus Demystified

CHAPTER 2
EXAMPLE 2.4 Foundations of Calculus 61
Discuss the limits of the function

2x − 4 if x< 2
f(x) = 2
x if x ≥ 2
at c = 2.
SOLUTION
As x approaches 2 from the left, f(x) = 2x − 4 approaches 0. As x
2
approaches 2 from the right, f(x) = x approaches 4. Thus we see that f
has left limit 0 at c = 2, written

lim f(x) = 0,
x→2 −
and f has right limit 4 at c = 2, written
lim f(x) = 4.
x→2 +
Note that the full limit lim x→2 f(x) does not exist (because the left and right
limits are unequal).

You Try It: Discuss one-sided limits at c = 3 for the function
 3
x − x if x< 3

f(x) = 24 if x = 3

4x + 1if x> 3

All the properties of limits that will be developed in this chapter, as well as the
rest of the book, apply equally well to one-sided limits as to two-sided (or standard)
limits.

2.2 Properties of Limits

To increase our facility in manipulating limits, we have certain arithmetical and
functional rules about limits. Any of these may be verified using the rigorous defi-
nition of limit that was provided at the beginning of the last section. We shall state
the rules and get right to the examples.
If f and g are two functions, c is a real number, and lim x→c f(x) and
lim x→c g(x) exist, then
Theorem 2.1
(a) lim x→c (f ± g)(x) = lim x→c f(x) ± lim x→c g(x);