AP Calc Notes: L-7 Relative rates/end behavior models
I could just go on and on about infinity…
Limits at Infinity
6 x4
.
x →∞ 2 x 2 + 1
Find the end behavior model for the function and then evaluate lim
6x 4
= lim 3x2 = ∞
x →∞ 2x2 + 1
x →∞
lim
The following facts are helpful when evaluating limits at ±∞:
1. For large x, a polynomial function behaves like its highest order term.
2. An exponential function ax (a > 1) grows faster than any power of x.
3. Any positive power of x grows faster than a log function logax (a > 1).
Comparing sizes – put in order from fastest growth on the top to slowest growth on the bottom
1
x, x 2 , x n , , ln x, e x , 2 x , x x , x !
x
Function
xx
x!
ex
2x
xn
x2
x
ln x
1
x
Limits of quotients of functions as x goes to infinity are comparing relative growth rates
faster
= ∞ or DNE
x →∞ slower
lim
slower
=0
x →∞ faster
lim
same
= ratio of coefficients
x →∞ same
lim
Evaluate the following
6x2
x →∞ 2 x 2 + 1
a. lim
6x2
= lim 3 = 3
2
x →∞ 2x
x →∞
= lim
3x − 2
x →∞ 4 x + 6 x + 1
b. lim
2
= lim
x →∞
3x
3
= lim
=0
2
x
→∞
4x
4x
x2
x →∞ 1010 x + 1
c. lim
x2
x
= lim 10 = lim 10 = ∞
x →∞ 10 x
x →∞ 10
d. lim
x →∞
3x 2 − 1
( 3x − 1)
2
3x2
3
1
= lim =
2
x →∞ 9x
x →∞ 9
3
= lim
e. lim
x →∞
4 x2 + 1
2x +1
= lim
x →∞
f. lim
x →−∞
4x2
| 2x |
= lim
= 1 (For x > 0, |2x| = 2x)
x →∞ 2x
2x
4 x2 + 1
2x +1
= lim
x →−∞
4x2
| 2x |
= lim
= -1 (For x < 0, |2x| = -2x)
x →−∞ 2x
2x
g. lim x 5e − x
x →∞
x5
= 0 since ex (an exponential) grows faster than x5 (a power).
x
x →∞ e
= lim
x1/ e
x →∞ ln x
h. lim
= ∞ since x1/e (a power) grows faster than ln x (a log function)
⎛π ⎞
i. lim cos ⎜ ⎟
x →∞
⎝x⎠
= cos( lim
x →∞
π
) = cos(0) = 1
x