1. Limits at Infinity and Horizontal Asymptotes
Lesson 2-6
Limits at Infinity and Horizontal Asymptotes

2. Objectives
Identify and use limits of functions as x approaches either +/- ∞
Identify horizontal asymptotes of functions

3. Vocabulary
Horizontal Asymptote – a line y = L is a horizontal asymptote, if either limx→∞ f(x) = L or limx→-∞ f(x) = L
Infinity – ∞ (not a number!! ∞ - ∞ ≠ 0)

4. Limits at Infinity Horizontal Asymptotes:
10x² + 9 f(x) = x² + 1
16x4 + x² g(x) = x4 + 7
y = 4
y = 2
x² (10 + 9/x²) lim f(x) = lim = lim = x² (5 + 1/x²)
x4 (16 + 1/x²) lim g(x) = lim = lim = x4 (4 + 7/x4)
x→∞
x→∞
x→∞
x→-∞
x→-∞
x→-∞
11x4 + x² h(x) = x2 + 7
x2 (11x2 + 1) x² lim h(x) = lim = lim = ∞ x2 (3 + 7/x2)
x→∞
x→∞
x→∞
lim (x² - 5x) = lim x² - 5 lim x = ∞ not ∞ - ∞ !!
x→∞
x→∞
x→∞
Remember infinity is not a number!

5. Rational Functions
When given a ratio of two polynomials, the limit of the function as x approaches infinity will be determined by the ratio of highest powers (HP) of x in numerator and the denominator:
HPs equal: then the limit is the ratio of the constants in front of the HP x-terms (and its horizontal asymptote)
HP in numerator > HP in denominator: then the limit is DNE (and no horizontal asymptotes exist)
HP in numerator < HP in denominator: then the limit is 0 (and the horizontal asymptote is y = 0)
7x³ - 3x² - 2x
example: lim =
x x³ - 13x²
5x³ + 7x² - 3x + 4
example: lim = DNE
x x² - 8x + 5
-6x² - 8x - 7
example: lim = 0
x x³ + 7

6. Horizontal Asymptotes
A horizontal asymptote for a function f is a line y = L such that , either , or , or both.
A function may have at most 2 horizontal asymptotes.
lim f(x) = L
x
lim f(x) = L
x-

7. Example 1 Evaluate: a. b. c. d. 5x³ + 7x + 1
lim =
x x³ + 2x² + 3
2x + 5
lim =
x  x² + 4
cos x
lim =
x x
x³ + 6x + 1
lim =
x x² - 5x

8. Example 2 Find the horizontal asymptote(s) for x² - 2x + 1
y = x³ + 4x x7 – 4x5 + 3x - 1
y = x x
y = x² - 1

9. Checking for Understanding

10. Summary & Homework Summary:
Limits at infinity involved the highest powers in the function
Horizontal asymptotes (y = L) are the limits that exist (as x approaches infinity)
Homework: pg : 2, 3, 7, 11, 13, 18, 27, 29, 33, 38, 39