In symbol, we write this as

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2.2 Limits Involving Infinity

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In symbol, we write this as LIMIT AT INFINITY Consider the function Investigate the behavior of f(x) as x decreases without bound x -1 -5 -10 -100 -1000 -10,000 f(x) 1 1.92 1.98 1.9998 1.999998 1.99999998 In symbol, we write this as

In symbol, we write this as Consider the function Investigate the behavior of f(x) as x increases without bound x 1 5 10 100 1000 10,000 f(x) 1.92 1.98 1.9998 1.999998 1.99999998 In symbol, we write this as

horizontal asymptote

Theorem : Let n be a positive integer, then

Horizontal asymptote

x = -2 (vertical asymptote) y = 2 is a horizontal asymptote Dh: all x R except x = -2 When x = 0 , y = - 3 (y – int) When y = 0 , x = 3 (x – int) x = -2 (vertical asymptote) y = 2 is a horizontal asymptote x - 4 - 3 - 1 y = h(x) 7 12 - 8 9

x = -2 (vertical asymptote) y = 2 is a horizontal asymptote Dh: all x R except x = -2 horizontal asymptote y = 2 x = -2 (vertical asymptote) x - 4 - 3 - 1 y = h(x) 7 12 - 8 (0, -3) , (3, 0) vertical asymptote x = -2 y = 2 is a horizontal asymptote 10

Df: all x R except x = -2 & x = 2 When x = 0 , (y – int) When y = 0 , x = -1 (x – int) x = -2 (vertical asymptote) x = 2 (vertical asymptote) x -4 -3 -1.5 1 3 4 f(x) (-1/4) (-2/5) (2/7) (-2/3) (4/5) (5/12) -0.25 -0.4 0.286 -0.67 0.8 0.417 y = 0 (horizontal asymptote) 11

Df: all x R except x = -2 & x = 2 x = -2 (vertical asymptote) vertical asymptote horizontal asymptote x = -2 y = 0 x = 2 (vertical asymptote) x = 2 vertical asymptote y = 0 (horizontal asymptote) x -4 -3 -1.5 1 3 4 f(x) (-1/4) (-2/5) (2/7) (-2/3) (4/5) (5/12) -0.25 -0.4 0.286 -0.67 0.8 0.417 12

x -1 1 f (x) e EVALUATING LIMITS OF EXPONENTIAL FUNCTIONS Using graph 1 f (x) e Using graph (1 , e) (0 , 1) 15

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EVALUATING LIMITS OF Logarithmic FUNCTIONS Using graph

EVALUATING LIMITS OF Logarithmic FUNCTIONS Using graph

EVALUATING LIMITS OF Trigonometric FUNCTIONS Using graph

EVALUATING LIMITS OF Trigonometric FUNCTIONS Using graph

EVALUATING LIMITS OF SPECIAL FUNCTIONS Using graph

EVALUATING LIMITS OF SPECIAL FUNCTIONS Using graph

EVALUATING LIMITS OF SPECIAL FUNCTIONS Using graph