Chapter 2 | Limits
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Theorem 2.7: The Squeeze Theorem Let f ( x ), g ( x ), and h ( x ) be defined for all x ≠ a over an open interval containing a . If f ( x ) ≤ g ( x ) ≤ h ( x ) for all x ≠ a in an open interval containing a and lim x → a f ( x ) = L = lim x → a h ( x ) where L is a real number, then lim x → a g ( x ) = L .
Example 2.24 Applying the Squeeze Theorem
Apply the squeeze theorem to evaluate lim x →0
x cos x .
Solution Because −1≤cos x ≤1 for all x , we have − | x | ≤ x cos x ≤ | x | . Since lim x →0
(− | x |) =0= lim x →0
| x |, fromthe
x cos x =0. The graphs of f ( x ) = − | x | , g ( x ) = x cos x , and h ( x ) = | x | are
squeeze theorem, we obtain lim x →0
shown in Figure 2.28 .
Figure 2.28 The graphs of f ( x ), g ( x ), and h ( x ) are shown around the point x =0.
2.19
x 2 sin 1 x .
Use the squeeze theorem to evaluate lim x →0
We now use the squeeze theorem to tackle several very important limits. Although this discussion is somewhat lengthy, these limits prove invaluable for the development of the material in both the next section and the next chapter. The first of these limits is lim θ →0 sin θ . Consider the unit circle shown in Figure 2.29 . In the figure, we see that sin θ is the y -coordinate on the unit circle and it corresponds to the line segment shown in blue. The radian measure of angle θ is the length of the arc it subtends on the unit circle. Therefore, we see that for 0< θ < π 2 , 0< sin θ < θ .
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