Limits at Infinity and Horizontal Asymptotes

1
Lesson 2-6

• Limits at Infinity and Horizontal Asymptotes

2
Objectives

• Identify and use limits of functions as x
  approaches either /- 8
• Identify horizontal asymptotes of functions

3
Vocabulary

• Horizontal Asymptote a line y L is a
  horizontal asymptote, if either limx?8 f(x) L
  or limx?-8 f(x) L
• Infinity 8 (not a number!! 8 - 8 ? 0)

4
Limits at Infinity
Horizontal Asymptotes
10x² 9f(x) -------------
5x² 1
16x4 x²g(x) -------------
4x4 7
y 4
y 2
x² (10 9/x²)
10 lim f(x) lim -------------------- lim
----- 2 x² (5 1/x²)
5
x4 (16 1/x²)
16 lim g(x) lim -------------------- lim
----- 4 x4 (4 7/x4)
4
x?8
x?-8
x?8
x?8
x?-8
x?-8
11x4 x²h(x) -------------
3x2 7
x2 (11x2 1)
11x² lim h(x) lim --------------------
lim ------- 8 x2 (3
7/x2) 3
x?8
x?8
x?8
lim (x² - 5x) lim x² - 5 lim x 8 not 8 - 8
!!
x?8
x?8
x?8
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 gt HP in denominator then the
  limit is DNE(and no horizontal asymptotes exist)
• HP in numerator lt HP in denominator then the
  limit is 0(and the horizontal asymptote is y 0)

7x³ - 3x² - 2x 1
7 example lim --------------------
------ ----
x?? 4x³ - 13x² 7
4
5x³ 7x² - 3x 4
example lim ---------------------
----- DNE
x?? 3x² - 8x 5

-6x² - 8x - 7
example lim ----------------------
0
x?? 2x³ 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??
3x³ 2x² 3
cos x lim ---------------
x?? x
x³ 6x 1 lim
--------------------- x??
2x² - 5x
8
Example 2

• Find the horizontal asymptote(s) for
• x² - 2x 1
• y ------------------- 3x³ 4x 7
  3x7 4x5 3x - 1
• y -------------------------- 2x7
  4 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 146 - 149 2, 3, 7, 11, 13, 18, 27,
  29, 33, 38, 39