Lesson 05 - limits involving infinity PDF

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Limits Involving Infinity

AP Calculus AB Unit 1 – Lesson 5 The problem with infinity is that even if we simplify what we are given, we can’t “plug in” infinity. First, when dealing with limits approaching infinity, look at the dominant terms. Example 1:

x 2  3x  2 x  x2  5 lim

x 2 3 x  2 x  x2  5 lim

Example 2:

x 2  3x  2 x  x 5 lim

x 2  3x  2 x  x 5 lim

Example 3:

lim x 

x 5 x  3x  2 2

x 5 x  x  3x  2 lim

Example 4:

lim

1 x7

lim

1 x7

x 7

x 7

Example 5:

2

lim

x x 5

lim

x x5

x 5 

x 5

If lim f  x   b , then y  b is a ________________________________ for the function. x 

If lim f  x   or lim f  x   , then x  a is a ________________________________ for the x a

x a

function.

Example 6: Find the horizontal asymptote(s) of f  x  

Example 7: Find the vertical asymptote(s) of f  x   right of each asymptote.

x x2 1

3 x 2 and determine the behavior on the left and x5

Find lim f x  , lim f  x , and any horizontal asymptotes of f  x  .   x

x

(Section 2.2: 1-6) 1. lim cos  1x   x

lim cos  1x  

x

2. lim

sin  2 x

x 

lim

x 

x



sin  2x  x



e x  x  x

3. lim

e x lim  x  x 3 x3  x  1  x  x 3

4. lim

3x3  x  1  x  x3 lim

3x 1  x  x  2

5. lim

3x 1  x  x  2 lim

e x 1   e x  3

6. lim x

ex  1  x  e x  3 lim

Limits Involving Infinity Summary Limits as x approaches infinity f  x f  x or lim , look at the dominant terms of f  x  and g  x  . When working with lim x  g x   x  g  x  This is side work. For example

x 2  3x  1 x 2 x2 , but we will look at to evaluate  2 2 2 4x  1 4x 4x

x 2  3x  1 . 2  4x  1

lim x

Good

Bad

lim

x 2  3x  1 1  x  4x2  1 4

1 x2  2 4x 4

x 2  3x  1 0 3 x  4x  1

x2 1 1   3 4x 4 x big

lim

1 x 2  3x  1 x 2  2 2 x  x x 4 1 4 4

lim

x 2  3x  1 x 2 1 1  3  0 3 x  4x  1 4x 4x 4   

lim

Horizontal Asymptotes To find horizontal asymptotes of f  x  , we must find lim f  x  and lim f  x  . If either of x 

x 

these is equal to a number b, then we have a horizontal asymptote at y  b. If neither is equal to a number, then the function has no horizontal asymptotes.

Limits that Result in an Answer of Infinity When working with limits like lim f  x  or lim f  x  , (mentally) pick a number close to a and x a

x a

then determine how this number would affect the function. 1 4 Examples: small small Vertical Asymptotes To find vertical asymptotes of f  x , we look for places where lim f  x    or x a

lim f  x    . Often, this happens where the denominator of f  x  is zero.

x a...