184
Chapter 2 | Limits
For what values of x is f ( x ) = x +1 x −5
continuous?
Solution The rational function f ( x ) = x +1 x −5
is continuous for every value of x except x =5.
For what values of x is f ( x ) =3 x 4 −4 x 2 continuous?
2.22
Types of Discontinuities As we have seen in Example 2.26 and Example 2.27 , discontinuities take on several different appearances. We classify the types of discontinuities we have seen thus far as removable discontinuities, infinite discontinuities, or jump discontinuities. Intuitively, a removable discontinuity is a discontinuity for which there is a hole in the graph, a jump discontinuity is a noninfinite discontinuity for which the sections of the function do not meet up, and an infinite discontinuity is a discontinuity located at a vertical asymptote. Figure 2.37 illustrates the differences in these types of discontinuities. Although these terms provide a handy way of describing three common types of discontinuities, keep in mind that not all discontinuities fit neatly into these categories.
Figure 2.37 Discontinuities are classified as (a) removable, (b) jump, or (c) infinite.
These three discontinuities are formally defined as follows:
Definition If f ( x ) is discontinuous at a , then 1. f has a removable discontinuity at a if lim x → a
f ( x ) exists. (Note: When we state that lim x → a
f ( x ) exists, we
f ( x ) = L , where L is a real number.)
mean that lim x → a
f ( x ) and lim x → a +
f ( x ) both exist, but lim x → a −
f ( x ) ≠ lim
f ( x ).
2. f has a jump discontinuity at a if lim x → a −
x → a +
f ( x ) and lim x → a +
f ( x ) both exist, we mean that both are real-valued and that
(Note: When we state that lim x → a −
neither take on the values ±∞.) 3. f has an infinite discontinuity at a if lim x → a −
f ( x ) =±∞ or lim x → a +
f ( x ) =±∞.
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