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
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
3x − 2
x →∞ 4 x + 6 x + 1
b. lim
2
x2
c. lim 10
x →∞ 10 x + 1
d. lim
x →∞
3x 2 − 1
( 3x − 1)
2
e. lim
4x2 + 1
2x +1
f. lim
4x2 + 1
2x +1
x →∞
x →−∞
g. lim x 5e − x
x →∞
x1/ e
h. lim
x →∞ ln x
π 
i. lim cos  
x →∞
x