408
Chapter 4 | Applications of Derivatives
1,000
10,000
10
100
x
2+ 1 x
2.01
2.001
2.0001
2.1
−10 −100 −1000 −10,000
x
2+ 1 x
1.9
1.99
1.999
1.9999
Table 4.2 Values of a function f as x →±∞
More generally, for any function f , we say the limit as x →∞ of f ( x ) is L if f ( x ) becomes arbitrarily close to L as long as x is sufficiently large. In that case, we write lim x →∞ f ( x ) = L . Similarly, we say the limit as x →−∞ of f ( x ) is L if f ( x ) becomes arbitrarily close to L as long as x <0 and | x | is sufficiently large. In that case, we write lim x →−∞ f ( x ) = L . We now look at the definition of a function having a limit at infinity. Definition (Informal) If the values of f ( x ) become arbitrarily close to L as x becomes sufficiently large, we say the function f has a limit at infinity and write lim x →∞ f ( x ) = L . If the values of f ( x ) becomes arbitrarily close to L for x <0 as | x | becomes sufficiently large, we say that the function f has a limit at negative infinity and write lim x →–∞ f ( x ) = L . If the values f ( x ) are getting arbitrarily close to some finite value L as x →∞ or x →−∞, the graph of f approaches the line y = L . In that case, the line y = L is a horizontal asymptote of f ( Figure 4.41 ). For example, for the function f ( x ) = 1 x , since lim x →∞ f ( x ) =0, the line y =0 is a horizontal asymptote of f ( x ) = 1 x .
Definition If lim x →∞
f ( x ) = L or lim x
→−∞ f ( x ) = L , we say the line y = L is a horizontal asymptote of f .
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