Page 71 - Calculus Demystified

Foundations of Calculus
CHAPTER 2
58 understanding of limit. We now develop that understanding with some carefully
chosen examples.
EXAMPLE 2.1

Define 3 − x if x< 1
f(x) = 2
x + 1 if x> 1
See Fig. 2.1. Calculate lim x→1 f(x).

Fig. 2.1
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SOLUTION
Observe that, when x is to the left of 1 and very near to 1 then f(x) = 3 − x
is very near to 2. Likewise, when x is to the right of 1 and very near to 1 then
2
f(x) = x + 1 is very near to 2. We conclude that
lim f(x) = 2.
x→1
Wehavesuccessfullycalculatedourfirstlimit.Figure2.1confirmstheconclusion
that our calculations derived.
EXAMPLE 2.2
Define
2
x − 4
g(x) = x − 2 .
Calculate lim x→2 g(x).
SOLUTION
We observe that both the numerator and the denominator of the fraction
defining g tend to 0 as x → 2 (i.e., as x tends to 2). Thus the question seems
to be indeterminate.
However, we may factor the numerator as x − 4 = (x − 2)(x + 2).
2
As long as x = 2 (and these are the only x that we examine when we

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