Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

Transcription

1 Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

2 Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

3 Infinite Limits Ex : Find,, and by x 0 + x x 0 x x 0 x a) making a table b) looking at a graph

4 Infinite Limits Ex : Find,, and by x 0 + x x 0 x x 0 x a) making a table x /x ? So x 0 + x = + and x 0 x = x 0 x = DNE

5 Infinite Limits Ex : Find,, and by x 0 + x x 0 x x 0 x b) looking at a graph x 0 + x = + x 0 x = x 0 x = DNE

6 Infinite Limits Ex 2: Find,, and by x 0 + x 2 a) making a table b) looking at a graph x 0 x 2 x 0 x 2

7 Infinite Limits Ex 2: Find,, and by x 0 + x 2 a) making a table x 0 x 2 x 0 x 2 x /x^2 00 0,000,000,000?,000,000 0, So x 0 + x 2 = + and x 0 x 2 = + x 0 x 2 = +

8 Infinite Limits Ex 2: Find,, and by x 0 + x 2 b) looking at a graph x 0 x 2 x 0 x 2 x 0 + x 2 = + x 0 x 2 = + x 0 x 2 = +

9 Informal Definition of an Infinite Limit ) f x = + means x a if you plug in numbers closer and closer to a (on either side of a but not a itself!) into the function, the outputs will get bigger and bigger without bound. 2) f x = + means x a if you plug in numbers closer and closer to a but less than a into the function, the outputs will get bigger and bigger without bound. 3) f x = + means + x a If you plug in numbers closer and closer to a but greater than a into the function, the outputs will get bigger and bigger without bound.

10 Informal Definition of an Infinite Limit ) f x = means x a if you plug in numbers closer and closer to a (on either side of a but not a itself!) into the function, the outputs will get smaller and smaller without bound. 2) f x = means x a if you plug in numbers closer and closer to a but less than a into the function, the outputs will get smaller and smaller without bound. 3) f x = means + x a If you plug in numbers closer and closer to a but greater than a into the function, the outputs will get smaller and smaller without bound.

11 Infinite Limits Notes: When the it is of the form nonzero, the answer for the 0 it will either be +,, or DNE. Q: How can you tell which one? A: Think about making a table and consider the sign of the outputs. ) If the outputs are always positive (eventually), the answer will be + 2) If the outputs are always negative (eventually), the answer will be 3) Otherwise the answer will be DNE

12 Infinite Limits Notes: In this class, if a one-sided it is of the form nonzero the answer for the it will either be + or (not DNE). The only time we ll see DNE is if the left and right hand its are different. If a it is + or, this doesn t mean that the it exists. The it doesn t exist because if a it exists, it must be a number. But by saying that a it is + or, you are explaining HOW it doesn t exist. But if a it is + or, do not write that the it DNE, just know that it doesn t exist. 0,

13 Infinite Limits Notes: Think of a it of the form nonzero 0 like this: When you divide (a nonzero number) by a really small number, you get a really big number (+ or )

14 Ex 3: Find x 4 2x 9 x 4 7 Infinite Limits

15 Infinite Limits Ex 4 (book sec. 2.2 hw #39): Find xcsc(x) x 2π

16 Infinite Limits Ex 5 (book sec. 2.2 hw #42): Find x 0 + x lnx

17 Infinite Limits Ex 6: Find x 2 lnx 2x

18 Vertical Asymptotes Q: What is a vertical asymptote?

19 Vertical Asymptotes Q: What is a vertical asymptote? vertical asymptotes x = a & x = b

20 Definition of a Vertical Asymptotes Def: The line x = a is a vertical asymptote of the function f if x a f x =, f x = +, x a f x = or f x = + x a + x a +

21 Vertical Asymptotes Steps in finding vertical asymptotes Find the points (a) of discontinuity of f Calculate f x and f x x a x a + If either of these answers are + or, then the line x = a is a vertical asymptote. Otherwise, the line x = a is not a vertical asymptote.

22 Vertical Asymptotes Ex 7: Find the vertical asymptotes of f x = x2 + 5x 4 a) x 2 b) g x = 7x + 0 sin (x) x c) h x = ln (x) d) k x = sin x e) L x = 2x 0 x 5

23 Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

24 Limits at Infinity Ex 8: Find x x by a) making a table b) looking at a graph

25 Limits at Infinity Ex 8: Find x x by a) making a table -,000,000-0, So x x = 0

26 Limits at Infinity Ex 8: Find x x by b) looking at the graph So x x = 0

27 Limits at Infinity Ex 9: Find x x 2 by a) Making a table b) Looking at a graph

28 Limits at Infinity Ex 9: Find x x 2 by a) Making a table 00 0,000,000, ^-8 0^-2 So x x 2 = 0

29 Limits at Infinity Ex 9: Find x x 2 by b) looking at the graph So x x 2 = 0

30 Limits at Infinity Ex 0: Use a table to find x 8x + 2x 5

31 Limits at Infinity Ex 0: Use a table to find x 8x + 2x

32 Limits at Infinity Ex 0: Use a table to find x 8x + 2x So guess is x 8x + 2x 5 = 4

33 Limits at Infinity Ex : Use the graph to find each of the following a) x tan (x) b) c) x ln (x) sin(x) x

34 Definition of a Limit at Infinity / Horizontal Asymptotes Let L be a real number and f be a function. ) f x = L means x if you plug in bigger and bigger (positive) numbers into the function, the outputs will get closer and closer to L. 2) If f x = L, then the line y = L is called a x horizontal asymptote of f.

35 Definition of a Limit at Infinity/ 3) f x = L means x Horizontal Asymptotes if you plug in smaller and smaller (or bigger and bigger negative) numbers into the function, the outputs will get closer and closer to L. 4) If f x = L, then the line y = L is called a x horizontal asymptote of f.

36 Definition of a Limit at Infinity/ Horizontal Asymptotes Steps in finding the horizontal asymptotes of a function Find f x. If this it exists and equals the y x number L, then the line = L is a horizontal asymptote of the function f. Find f x. If this it exists and equals the y x number R, then the line = R is a horizontal asymptote of the function f.

37 Definition of a Limit at Infinity/ Horizontal Asymptotes Notes: A function can have 0,, or 2 horizontal asymptotes Rational expressions can have 0 or horizontal asymptotes, but not 2 (as you ll see shortly) When finding the horizontal asymptotes of algebraic expressions, be careful with your algebra.

38 Result: ) If r > 0 is a rational number, then x x r = 0 2) If r > 0 is a rational number such that is defined for all x, then x x r = 0 x r

39 Horizontal Asymptotes Ex 2: Find the horizontal asymptotes of a) f x = x + 3 x 2 7x + 4 b) g x = 5x3 + 4x 9 2x 3 + x + c) h x = 2x3 + 3 x 2

40 Result on Horizontal Asymptotes of Rational Expressions If the degree top < degree bottom, the graph only has one horizontal asymptote. The horizontal asymptote is y = 0 (i.e. the x-axis) If the degree top = degree bottom, the graph only has one horizontal asymptote. The horizontal asymptote is lead coefficient of top y = lead coefficient of bottom If the degree top > degree bottom, the graph has no horizontal asymptote

41 Horizontal Asymptotes Ex 3: Find the horizontal asymptotes of a) f x = 5x 3 4x 6 + 2x 4 3 b) g x = 27x6 4x x 3 8x

42 Limits at Infinity Ex 4: Find x 4x2 + 3x + 2x