Chapter 4 | Applications of Derivatives
415
for all x > N . Similarly we can define limits as x →−∞.
Figure 4.49 For a function with an infinite limit at infinity, for all x > N , f ( x ) > M .
Earlier, we used graphical evidence ( Figure 4.47 ) and numerical evidence ( Table 4.3 ) to conclude that lim x →∞ x 3 =∞. Here we use the formal definition of infinite limit at infinity to prove that result. Example 4.23 An Infinite Limit at Infinity
x 3 =∞.
Use the formal definition of infinite limit at infinity to prove that lim x →∞
Solution Let M >0. Let N = M 3 .
Then, for all x > N , we have
⎛ ⎝ M 3
⎞ ⎠
3
x 3 > N 3 =
= M .
x 3 =∞.
Therefore, lim x →∞
x 2 =∞.
4.22
Use the formal definition of infinite limit at infinity to prove that lim x →∞ 3
End Behavior The behavior of a function as x →±∞ is called the function’s end behavior . At each of the function’s ends, the function could exhibit one of the following types of behavior: 1. The function f ( x ) approaches a horizontal asymptote y = L . 2. The function f ( x )→∞ or f ( x )→−∞. 3. The function does not approach a finite limit, nor does it approach ∞ or −∞. In this case, the function may have some oscillatory behavior.
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