Transcript Lesson 6-2

Lesson 2-6
Limits at Infinity
and Horizontal Asymptotes
Objectives
• Identify and use limits of functions as x
approaches either +/- ∞
• Identify horizontal asymptotes of functions
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)
Horizontal Asymptotes:
Limits at Infinity
16x4 + x²
g(x) = ------------4x4 + 7
10x² + 9
f(x) = ------------5x² + 1
y=4
y=2
x4 (16 + 1/x²)
16
lim g(x) = lim -------------------- = lim ----- = 4
x→-∞ 4
x→-∞
x→-∞ x4 (4 + 7/x4)
x² (10 + 9/x²)
10
lim f(x) = lim -------------------- = lim ----- = 2
x→∞ 5
x→∞
x→∞ x² (5 + 1/x²)
11x4 + x²
h(x) = ------------3x2 + 7
x2 (11x2 + 1)
11x²
lim h(x) = lim -------------------- = lim ------- = ∞
x→∞
x→∞
x→∞ x2 (3 + 7/x2)
3
lim (x² - 5x) = lim x² - 5 lim x = ∞ not ∞ - ∞ !!
x→∞
x→∞
x→∞
Remember infinity is not a number!
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:
1) HPs equal: then the limit is the ratio of the constants in front of
the HP x-terms (and its horizontal asymptote)
7x³ - 3x² - 2x + 1
example: lim -------------------------x
4x³ - 13x² + 7
=
7
---4
2) HP in numerator > HP in denominator: then the limit is DNE
(and no horizontal asymptotes exist)
5x³ + 7x² - 3x + 4
example: lim -------------------------x
3x² - 8x + 5
= DNE
3) HP in numerator < HP in denominator: then the limit is 0
(and the horizontal asymptote is y = 0)
-6x² - 8x - 7
example: lim ---------------------x
2x³ + 7
= 0
Horizontal Asymptotes
A horizontal asymptote for a function f is a line y = L such that ,
either lim f(x) = L , or lim f(x) = L , or both.
x
x-
A function may have at most 2 horizontal asymptotes.
5x 3  7 x  1
x  3 x 3  2 x 2  3
lim
Example 1
Evaluate:
a.
lim
x
5x³ + 7x + 1
----------------------3x³ + 2x² + 3
b.
2x + 5
lim --------------x
 x² + 4
=
c.
cos x
lim --------------x
x
=
d.
lim
x
x³ + 6x + 1
--------------------2x² - 5x
=
=
Example 2
Find the horizontal asymptote(s) for
x² - 2x + 1
a. y = ------------------3x³ + 4x + 7
3x7 – 4x5 + 3x - 1
b. y = -------------------------2x7 + 4
x
c. y = ---------------x² - 1
Checking for Understanding
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 146 - 149: 2, 3, 7, 11, 13,
18, 27, 29, 33, 38, 39