Chapter 2 | Limits
145
Based on this table, we can conclude that a. lim x →2 − f ( x ) =0. Therefore, the (two-sided) limit of f ( x ) does not exist at x =2. Figure 2.18 shows a graph of f ( x ) and reinforces our conclusion about these limits. f ( x ) =3 andb. lim x →2 +
Figure 2.18 The graph of f ( x ) = ⎧ ⎩
⎨ x +1 if x <2 x 2 −4 if x ≥2
has a
break at x =2.
2.7
Use a table of functional values to estimate the following limits, if possible.
| x 2 −4 | x −2 | x 2 −4 | x −2
lim x →2 −
a.
lim x →2 +
b.
Let us now consider the relationship between the limit of a function at a point and the limits from the right and left at that point. It seems clear that if the limit from the right and the limit from the left have a common value, then that common value is the limit of the function at that point. Similarly, if the limit from the left and the limit from the right take on different values, the limit of the function does not exist. These conclusions are summarized in Relating One-Sided and Two- Sided Limits . Theorem 2.2: Relating One-Sided and Two-Sided Limits Let f ( x ) be a function defined at all values in an open interval containing a , with the possible exception of a itself, and let L be a real number. Then, lim x → a f ( x ) = L .if and only if lim x → a − f ( x ) = L and lim x → a + f ( x ) = L .
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