Sec 2 6 Limits Involving Infinity Asymptotes of

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Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Does not exist number

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Does not exist number

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: Consider the function: Study

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: Consider the function: Study the behavior of the function around x=0. 0. 1 10 0. 01 100 0. 001 1000 0. 0001 10000 0. 000001 1000000 Vertical Asymptote

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: 1. 1 10 1.

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: 1. 1 10 1. 01 100 1. 001 1000 0. 9 -10 1. 0001 10000 0. 999 -1000 1. 000001 1000000 0. 99999 -10000 0. 999999 -1000000 Vertical Asymptote

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Infinite Limits

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Infinite Limits

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Vertical Asymptote &

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Vertical Asymptote &

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Find all vertical asymptotes

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Find all vertical asymptotes

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs EXAMPLE

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs EXAMPLE

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Does not exist number

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Does not exist number

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: 1 1 10 0.

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: 1 1 10 0. 01 100 0. 0001 1000 0. 000001 ------ 1, 000 0. 000001 1, 000 10^(-12)

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Limits at Infinity of Rational

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Limits at Infinity of Rational Functions Example: To evaluate the limit at infinity of any rational function, we first divide both the numerator and denominator by the highest power of that occurs in the denominator.

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Remark: To evaluate the limit

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Remark: To evaluate the limit at infinity of any rational function, we first divide both the numerator and denominator by the highest power of that occurs in the denominator. Example: Remark:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Notes:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Notes:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Notes:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Notes:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: (ex 80 p 116)

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: (ex 80 p 116) Multiply by conjugate radical. Example:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Vertical Asymptote & Horizontal Asymptote

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Vertical Asymptote & Horizontal Asymptote & The line Is a horizontal asymptote

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: The line Is a

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: The line Is a horizontal asymptote

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: The line Is a

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: The line Is a horizontal asymptote

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs EXAM-1 TERM-121

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs EXAM-1 TERM-121

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Sec 2. 2

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Sec 2. 2

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs To evaluate the limit at

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs To evaluate the limit at infinity of any rational function, we first divide both the numerator and denominator by the highest power of that occurs in the denominator. Multiply by conjugate radical. Factor then take the limit

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Vertical Asymptote Horizontal Asymptote Oblique

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Vertical Asymptote Horizontal Asymptote Oblique Asymptote If the degree of the numerator of a rational function is 1 greater than the degree of the denominator, the graph has an oblique or slant line asymptote. We find an equation for the asymptote by dividing numerator by denominator Remark:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Who is going faster to

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Who is going faster to infinity Example:

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: Sketch the graph of

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Example: Sketch the graph of

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Exam 1 Term 101

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Exam 1 Term 101

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs a b

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs a b

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Find Do you have the

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Find Do you have the graph From the graph study the limit from right and left Does f contain greatest integer or absolute value Study the limit from right and left Substitute and find the limit Can we use Direct substitution Use: 1)factor then cancel 2)Multiply by conjugate 3)Make common denominator Write f as: Use squeeze theorem

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Sandwich thm Factor then take

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Sandwich thm Factor then take the limit high pwr in denomi Rational func Containing radicals Containing absolute value Remove the | | Multiply by conjugate radical. Containing noninteger Use graph Use

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Reminder: After sec 2. 5

Sec 2. 6: Limits Involving Infinity; Asymptotes of Graphs Reminder: After sec 2. 5 continuity