Chapter 2 | Limits
187
(−∞, −2), (−2, 0), and (0, +∞).
Example 2.34 Continuity over an Interval
State the interval(s) over which the function f ( x ) = 4− x 2 is continuous.
Solution From the limit laws, we know that lim x
x 2 = 4− a 2 for all values of a in (−2, 2). We also know that
→ a 4−
4− x 2 =0 exists and lim x →2 −
4− x 2 =0 exists. Therefore, f ( x ) is continuous over the interval
lim x →−2 + [−2, 2].
2.24
State the interval(s) over which the function f ( x ) = x +3 is continuous.
The Composite Function Theorem allows us to expand our ability to compute limits. In particular, this theorem ultimately allows us to demonstrate that trigonometric functions are continuous over their domains.
Theorem 2.9: Composite Function Theorem If f ( x ) is continuous at L and lim x → a g ( x ) = L , then lim x → a f ⎛ ⎝ g ( x ) ⎞ ⎠ = f ⎛ ⎝ lim x → a g ( x ) ⎞
⎠ = f ( L ).
Before we move on to Example 2.35 , recall that earlier, in the section on limit laws, we showed lim x →0 cos x = 1 = cos(0). Consequently, we know that f ( x ) =cos x is continuous at 0. In Example 2.35 we see how to combine this result with the composite function theorem. Example 2.35 Limit of a Composite Cosine Function
⎛ ⎝ x − π 2
⎞ ⎠ .
Evaluate lim
cos
x → π /2
Solution
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