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Chapter 2 | Limits
Example 2.27 Determining Continuity at a Point, Condition 2 Using the definition, determine whether the function f ( x ) = ⎧ ⎩
⎨ − x 2 +4 if x ≤3 4 x −8 if x >3
is continuous at x =3. Justify
the conclusion.
Solution Let’s begin by trying to calculate f (3).
f (3) = −(3 2 )+4=−5.
Thus, f (3) is defined. Next, we calculate lim x →3
f ( x ). To do this, we must compute lim x →3 −
f ( x ) and
f ( x ):
lim x →3 +
f ( x ) = −(3 2 )+4=−5
lim x →3 −
and
f ( x )=4(3)−8=4.
lim x →3 +
f ( x ) does not exist. Thus, f ( x ) is not continuous at 3. The graph of f ( x ) is shown in Figure
Therefore, lim x →3
2.36 .
Figure 2.36 The function f ( x ) is not continuous at 3 because lim x →3 f ( x ) does not exist.
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