168
Chapter 2 | Limits
lim x →3 +
x −3
b.
Solution Figure 2.25 illustrates the function f ( x ) = x −3 and aids in our understanding of these limits.
Figure 2.25 The graph shows the function f ( x ) = x −3. a. The function f ( x ) = x −3 is defined over the interval [3, +∞). Since this function is not defined to the left of 3, we cannot apply the limit laws to compute lim x →3 − x −3. In fact, since f ( x ) = x −3 is undefined to the left of 3, lim x →3 − x −3 does not exist. b. Since f ( x ) = x −3 is defined to the right of 3, the limit laws do apply to lim x →3 + x −3. By applying these limit laws we obtain lim x →3 + x −3=0.
In Example 2.22 we look at one-sided limits of a piecewise-defined function and use these limits to draw a conclusion about a two-sided limit of the same function. Example 2.22 Evaluating a Two-Sided Limit Using the Limit Laws
⎧ ⎩
⎨ 4 x −3 if x <2 ( x −3) 2 if x ≥2
For f ( x ) =
, evaluate each of the following limits:
f ( x )
lim x →2 − lim x →2 +
a.
f ( x )
b.
f ( x )
c. lim x →2
Solution Figure 2.26 illustrates the function f ( x ) and aids in our understanding of these limits.
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