James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

appendix F Proofs of Theorems A43
mapped into the interval sx 0 2 «, x 0 1 «d by f 21 . (See the arrow diagram in Figure 1.)
We have therefore found a number . 0 such that
f(x¸-∑)
{
if | y 2 y 0 | , then | f 21 syd 2 f 21 sy 0 d | , « x
y¸
∂¡ ∂
f(x¸+∑)
}
y
f
f–!
f
FIGURE 1
{ { }
}
a x¸-∑
x¸
x¸+∑
b
This shows that lim y l y 0
f 21 syd − f 21 sy 0 d and so f 21 is continuous at any number y 0 in
its domain.
■
8 Theorem If f is continuous at b and lim x l a tsxd − b, then
lim f stsxdd − f sbd
x la
ProoF Let « . 0 be given. We want to find a number . 0 such that
if 0 , | x 2 a | , then | f stsxdd 2 f sbd | , «
Since f is continuous at b, we have
and so there exists 1 . 0 such that
lim f syd − f sbd
y l b
if 0 , | y 2 b | , 1 then | f syd 2 f sbd | , «
Since lim x l a tsxd − b, there exists . 0 such that
if 0 , | x 2 a | , then | tsxd 2 b | , 1
Combining these two statements, we see that whenever 0 , | x 2 a | , we have
| tsxd 2 b | , 1, which implies that | f stsxdd 2 f sbd | , «. Therefore we have proved
that lim x l a f stsxdd − f sbd.
■
Section 3.3
sin
The proof of the following result was promised when we proved that lim − 1.
l 0
Theorem If 0 , , y2, then < tan .
ProoF Figure 2 shows a sector of a circle with center O, central angle , and radius 1.
Then
| AD | − | OA | tan − tan
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