Chapter 2 | Limits
185
Example 2.30 Classifying a Discontinuity
2 −4
In Example 2.26 , we showed that f ( x ) = x
is discontinuous at x =2. Classify this discontinuity as
x −2
removable, jump, or infinite.
Solution To classify the discontinuity at 2 we must evaluate lim x →2 f ( x ): lim x →2 f ( x ) = lim x →2 x 2 −4 x −2 = lim x →2
( x −2)( x +2) x −2
( x +2)
= lim
x →2
=4.
f ( x ) exists, f has a removable discontinuity at x =2.
Since f is discontinuous at 2 and lim x →2
Example 2.31 Classifying a Discontinuity
In Example 2.27 , we showed that f ( x ) = ⎧ ⎩ discontinuity as removable, jump, or infinite.
⎨ − x 2 +4 if x ≤3 4 x −8 if x >3
is discontinuous at x =3. Classify this
Solution Earlier, we showed that f is discontinuous at 3 because lim x →3
f ( x ) does not exist. However, since
f ( x ) =−5 and lim x →3 +
f ( x ) =4 both exist, we conclude that the function has a jump discontinuity at 3.
lim x →3 −
Example 2.32 Classifying a Discontinuity Determine whether f ( x ) = x +2 x +1
is continuous at −1. If the function is discontinuous at −1, classify the
discontinuity as removable, jump, or infinite.
Solution The function value f (−1) is undefined. Therefore, the function is not continuous at −1. To determine the type of
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