Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Slides:

Advertisements

Similar presentations

Rational Expressions GRAPHING.

Advertisements

Graphing Rational Functions

2.7 Rational Functions and Their Graphs Graphing Rational Functions.

Math 139 – The Graph of a Rational Function 3 examples General Steps to Graph a Rational Function 1) Factor the numerator and the denominator 2) State.

MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag Example: ,000,000 Example:

3.4 Rational Functions and Their Graphs

4.4 Rational Functions Objectives:

Infinite Limits and Limits to Infinity: Horizontal and Vertical Asymptotes.

3.7 Graphing Rational Functions Obj: graph rational functions with asymptotes and holes and evaluate limits of rational functions.

2.6 Rational Functions & Their Graphs

From sec 2.1 : Find: OUR GOAL Sec 2.2: Limit of Function and Limit Laws Sec 2.3: CALCULATING LIMITS USING THE LIMIT LAWS.

Limits at Infinity Explore the End Behavior of a function.

Limits at Infinity Horizontal Asymptotes Calculus 3.5.

MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag Example: ,000,000 Example:

AP CALCULUS AB Chapter 2: Limits and Continuity Section 2.2: Limits Involving Infinity.

2.7 Limits involving infinity Wed Sept 16

Rational Functions - Rational functions are quotients of polynomial functions: where P(x) and Q(x) are polynomial functions and Q(x)  0. -The domain of.

Introducing Oblique Asymptotes Horizontal Asymptote Rules: – If numerator and denominator have equal highest power, simplified fraction is the H.A. – If.

Class Work Find the real zeros by factoring. P(x) = x4 – 2x3 – 8x + 16

Rational Functions and Their Graphs. Example Find the Domain of this Function. Solution: The domain of this function is the set of all real numbers not.

Section 2.7. Graphs of Rational Functions Slant/Oblique Asymptote: in order for a function to have a slant asymptote the degree of the numerator must.

1 What you will learn 1. How to graph a rational function based on the parent graph. 2. How to find the horizontal, vertical and slant asymptotes for a.

1.5 Infinite Limits Objectives: -Students will determine infinite limits from the left and from the right -Students will find and sketch the vertical asymptotes.

2.6 (Day One) Rational Functions & Their Graphs Objectives for 2.6 –Find domain of rational functions. –Identify vertical asymptotes. –Identify horizontal.

L IMITS AND L IMITS AT INFINITY Limit Review 1. Limits can be calculated 3 ways Numerically Graphically Analytically (direct substitution) Properties.

Properties of Rational Functions 1. Learning Objectives 2 1. Find the domain of a rational function 2. Find the vertical asymptotes of a rational function.

2.2 Limits Involving Infinity Goals: Use a table to find limits to infinity, use the sandwich theorem, use graphs to determine limits to infinity, find.

As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: or.

1 Limits at Infinity Section Horizontal Asymptotes The line y = L is a horizontal asymptote of the graph of f if.

2.2 Limits Involving Infinity Greg Kelly, Hanford High School, Richland, Washington.

Symmetry and Asymptotes. f(-x) = f(x)EvenSymmetrical wrt y-axis f(-x) = -f(x)OddSymmetrical wrt origin Even Neither Odd Even Odd.

Limits at Infinity: End behavior of a Function

2.1 Rates of Change & Limits 2.2 Limits involving Infinity Intuitive Discussion of Limit Properties Behavior of Infinite Limits Infinite Limits & Graphs.

Limits Involving Infinity Section 1.4. Infinite Limits A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite.

Graphing Rational Functions Day 3. Graph with 2 Vertical Asymptotes Step 1Factor:

Bell Ringer. ASYMPTOTES AND GRAPHING December 2, 2015.

Rational Functions A rational function has the form

Ch. 2 – Limits and Continuity

Graphing Rational Functions Part 2

Polynomial and Rational Functions

Section 2.7B Slant Asymptotes

Rational Functions (Algebraic Fractions)

Rational functions are quotients of polynomial functions.

Ch. 2 – Limits and Continuity

Lesson 11.4 Limits at Infinity

OTHER RATIONAL FUNCTIONS

26 – Limits and Continuity II – Day 2 No Calculator

Graphing Polynomial Functions

3.5: ASYMPTOTES.

Let’s go back in time …. Unit 3: Derivative Applications

Objective: Section 3-7 Graphs of Rational Functions

Asymptotes Rise Their Lovely Heads

OUR GOAL Find: Sec 2.2: Limit of Function and Limit Laws

2.2 Limits at Infinity: Horizontal Asymptotes

Sec. 2.2: Limits Involving Infinity

Sec 4.5: Curve Sketching Asymptotes Horizontal Vertical

Limits at Infinity 3.5 On the agenda:

Graphing Rational Functions

2.2 Limits Involving Infinity

5-Minute Check Lesson 3-7.

Asymptotes Horizontal Asymptotes Vertical Asymptotes

2.6 Rational Functions and Their Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Properties of Rational Functions

2.5 Limits Involving Infinity

Limits at Infinity 4.5.

Asymptotes, End Behavior, and Infinite Limits

Limits Involving Infinity

Limits Involving Infinity

Presentation transcript:

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Does not exist number number

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Example: Consider the function: Study the behavior of the function around x=0. 10 0.1 100 0.01 1000 0.001 10000 0.0001 1000000 0.000001 Vertical Asymptote

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Example: 10 1.1 100 1.01 1000 1.001 10000 1.0001 1000000 1.000001 -10 0.9 -1000 0.999 -10000 0.99999 -1000000 0.999999 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 Infinite Limits

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 Find all vertical asymptotes

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs EXAMPLE

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Does not exist number number

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Example: Example: 1 0.1 10 0.01 100 0.001 1000 ------ ----- 0.000001 1,000,000 1 0.01 10 0.0001 100 0.000001 1000 ------ ----- 10^(-12) 1,000,000

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Example: Example: Example: Example:

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Limits at Infinity of Rational Functions 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: Example:

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: Example: Example: Remark:

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Notes: Notes:

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Notes: Notes:

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Example: (ex80p116) Multiply by conjugate radical. Example: Example:

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Example: Example: Example:

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 horizontal asymptote

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 EXAM-1 TERM-121

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs

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 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 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

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Exam1-Term101

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs b

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Limit Laws

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 Do you have the graph Find 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 Factor then take the limit Sandwich thm high pwr in denomi Rational func Containing radicals Remove the | | Containing absolute value Multiply by conjugate radical. Containing noninteger Use Use graph Use

Sec 2.6: Limits Involving Infinity; Asymptotes of Graphs Reminder: After sec2.5 continuity