188
Chapter 2 | Limits
⎛ ⎝ x − π 2
⎞ ⎠ =0 and cos x is continuous at 0,
The given function is a composite of cos x and x − π 2 .
Since lim
x → π /2
we may apply the composite function theorem. Thus, lim x → π /2 cos ⎛ ⎝ x − π 2 ⎞ ⎠ =cos ⎛ ⎝ lim x → π /2 ⎛ ⎝ x − π 2
⎞ ⎠ ⎞ ⎠ = cos(0) = 1.
x − π ).
Evaluate lim x
→ π sin(
2.25
The proof of the next theorem uses the composite function theorem as well as the continuity of f ( x ) = sin x and g ( x ) =cos x at the point 0 to show that trigonometric functions are continuous over their entire domains.
Theorem 2.10: Continuity of Trigonometric Functions Trigonometric functions are continuous over their entire domains.
Proof We begin by demonstrating that cos x is continuous at every real number. To do this, we must show that lim x → a cos x =cos a for all values of a . lim x → a cos x = lim x → a cos(( x − a )+ a ) rewrite x = x − a + a = lim x → a ⎛ ⎝ cos( x − a )cos a −sin( x − a )sin a ⎞ ⎠ apply the identity for the cosine of the sum of two angles
⎛ ⎝ lim x → a (
⎞ ⎠ cos a −sin ⎛
⎞ ⎠ sin a
x − a )
⎝ lim x → a (
x − a )
lim x → a ( x − a ) = 0, and sin x andcos x are continuous at 0
=cos
= cos(0)cos a − sin(0)sin a
evaluate cos(0) and sin(0) and simplify
=1·cos a −0· sin a =cos a . The proof that sin x is continuous at every real number is analogous. Because the remaining trigonometric functions may be expressed in terms of sin x and cos x , their continuity follows from the quotient limit law. □ As you can see, the composite function theorem is invaluable in demonstrating the continuity of trigonometric functions. As we continue our study of calculus, we revisit this theorem many times. The Intermediate Value Theorem Functions that are continuous over intervals of the form ⎡ ⎣ a , b ⎤ ⎦ , where a and b are real numbers, exhibit many useful properties. Throughout our study of calculus, we will encounter many powerful theorems concerning such functions. The first of these theorems is the Intermediate Value Theorem .
Theorem 2.11: The Intermediate Value Theorem Let f be continuous over a closed, bounded interval ⎡ ⎣ a , b ⎤
⎦ . If z is any real number between f ( a ) and f ( b ), then there
is a number c in ⎡
⎤ ⎦ satisfying f ( c ) = z in Figure 2.38 .
⎣ a , b
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