AP Calculus. Limits at Infinity and Asymptotes. Limits at Infinity. Horizontal
Asymptote. Limits at Infinity. 1) Graphically. 2) Numerically. -∞ . . . -100. -10. -1. 0
1. 10.
AP Calculus
Limits at Infinity and Asymptotes
Limits at Infinity
1) Graphically
2) Numerically
Horizontal Asymptote
-∞
...
-100
-10
-1
0
1
10
3
3
Example:
Numerator and Denominator both increase without bound as x increases towards ∞. (Indeterminate Form) CHANGE EQUATION
Divide by largest power of x.
Determining Limits at Infinity for Rational Functions
Example: (2 horizontal asymptotes)
Divide by x2
Divide by x2
Divide by x3
Largest power in denominator
Same power in numerator and denominator
Largest power in numerator
Lim = 0
Lim = ratio of leading coefficients
Lim DNE (unbounded behavior)
Horizontal Asymptote at y=0
Horizontal Asymptote at y = 2/3
No horizontal asymptote
With rational functions the horizontal asymptote is the same to the right and to the left. This is NOT true for other types of functions, for example, RADICAL functions.
