144
Chapter 2 | Limits
Limit from the left: Let f ( x ) be a function defined at all values in an open interval of the form ( c , a ), and let L be a real number. If the values of the function f ( x ) approach the real number L as the values of x (where x < a ) approach the number a , then we say that L is the limit of f ( x ) as x approaches a from the left. Symbolically, we express this idea as (2.6) lim x → a − f ( x ) = L . Limit from the right: Let f ( x ) be a function defined at all values in an open interval of the form ( a , c ), and let L be a real number. If the values of the function f ( x ) approach the real number L as the values of x (where x > a ) approach the number a , then we say that L is the limit of f ( x ) as x approaches a from the right. Symbolically, we express this idea as (2.7) lim x → a + f ( x ) = L .
Example 2.8 Evaluating One-Sided Limits
For the function f ( x ) = ⎧ ⎩
⎨ x +1 if x <2 x 2 −4 if x ≥2
, evaluate each of the following limits.
f ( x )
lim x →2 − lim x →2 +
a.
f ( x )
b.
Solution We can use tables of functional values again Table 2.6 . Observe that for values of x less than 2, we use f ( x ) = x +1 and for values of x greater than 2, we use f ( x ) = x 2 −4.
f ( x ) = x 2 −4
f ( x ) = x +1
x
x
1.9
2.9
2.1
0.41
1.99
2.99
2.01
0.0401
1.999
2.999
2.001
0.004001
1.9999
2.9999
2.0001
0.00040001
1.99999 2.99999
2.00001 0.0000400001
Table 2.6 Table of Functional Values for f ( x ) =
⎧ ⎩
⎨ x +1 if x <2 x 2 −4 if x ≥2
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