Limits at infinity (3.5) December 20th, 2011. I. limits at infinity Def. of Limit at Infinity: Let L be a real number. 1. The statement means that for.

Slides:

Advertisements

Similar presentations

9.3 Rational Functions and Their Graphs

Advertisements

3.5 Limits at Infinity Determine limits at infinity

Chapter 3: Applications of Differentiation L3.5 Limits at Infinity.

Miss Battaglia AP Calculus AB/BC. x -∞  ∞∞ f(x) 33 33 f(x) approaches 3 x decreases without bound x increases.

Functions AII.7 e Objectives: Find the Vertical Asymptotes Find the Horizontal Asymptotes.

Rational Functions A rational function is a function of the form where g (x) 0.

Ch. 9.3 Rational Functions and Their Graphs

1.6 Limits involving infinity. Infinite limits Definition: The notation (read as “the limit of of f(x), as x approaches a, is infinity”) means that the.

Section 5.2 – Properties of Rational Functions

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.

ACT Class Openers:

Rational Functions. 5 values to consider 1)Domain 2)Horizontal Asymptotes 3)Vertical Asymptotes 4)Holes 5)Zeros.

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

Ms. Battaglia AB/BC Calculus. Let f be the function given by 3/(x-2) A limit in which f(x) increases or decreases without bound as x approaches c is called.

9.3 Rational Functions and Their Graphs Rational Function – A function that is written as, where P(x) and Q(x) are polynomial functions. The domain of.

Infinite Limits Determine infinite limits from the left and from the right. Find and sketch the vertical asymptotes of the graph of a function.

2.2: LIMITS INVOLVING INFINITY Objectives: Students will be able to evaluate limits as Students will be able to find horizontal and vertical asymptotes.

Limits at Infinity Horizontal Asymptotes Calculus 3.5.

NPR1 Section 3.5 Limits at Infinity NPR2 Discuss “end behavior” of a function on an interval Discuss “end behavior” of a function on an interval Graph:

2.7 Limits involving infinity Wed Sept 16

2.6 Rational Functions and Asymptotes 2.7 Graphs of Rational Functions Rational function – a fraction where the numerator and denominator are polynomials.

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

Chapter 7 Polynomial and Rational Functions with Applications Section 7.2.

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.

Chapter Three: Section Five Limits at Infinity. Chapter Three: Section Five We have discussed in the past the idea of functions having a finite limit.

Warm-up Check skills p 491 (1 – 9). Section 9-3: Rational Functions and Their Graphs Goal 2.05: Use rational equations to solve problems. B) Interpret.

Sec 1.5 Limits at Infinity Divide numerator and denominator by the largest power of x in the denominator. See anything? f(x) has a horizontal Asymptote.

Calculus Chapter One Sec 1.5 Infinite Limits. Sec 1.5 Up until now, we have been looking at limits where x approaches a regular, finite number. But x.

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

LIMITS AT INFINITY Section 3.5.

Let’s develop a simple method to find infinite limits and horizontal asymptotes. Here are 3 functions that appear to look almost the same, but there are.

Limits, Asymptotes, and Continuity Ex.. Def. A horizontal asymptote of f (x) occurs at y = L if or Def. A vertical asymptote of f (x) occurs at.

1.5 Infinite Limits Chapter 1 – Larson- revised 9/12.

Aim: How do find the limit associated with horizontal asymptote? Do Now: 1.Sketch f(x) 2.write the equation of the vertical asymptotes.

Calculus Section 2.5 Find infinite limits of functions Given the function f(x) = Find =  Note: The line x = 0 is a vertical asymptote.

Lesson 3.5 Limits at Infinity. From the graph or table, we could conclude that f(x) → 2 as x → Graph What is the end behavior of f(x)? Limit notation:

Section 5: Limits at Infinity. Limits At Infinity Calculus

Rational Functions Objective: Finding the domain of a rational function and finding asymptotes.

Lines that a function approaches but does NOT actually touch.

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.

Miss Battaglia AP Calculus AB/BC. x -∞  ∞∞ f(x) 33 33 f(x) approaches 3 x decreases without bound x increases.

Infinite Limits Unit IB Day 5. Do Now For which values of x is f(x) = (x – 3)/(x 2 – 9) undefined? Are these removable or nonremovable discontinuities?

Calculus Section 2.5 Find infinite limits of functions Given the function f(x) = Find =  Note: The line x = 0 is a vertical asymptote.

Rational Functions…… and their Graphs

limits at infinity (3.5) September 15th, 2016

Rational Functions and Models

Ch. 2 – Limits and Continuity

Horizontal Asymptotes

8.2 Rational Functions and Their Graphs

Ch. 2 – Limits and Continuity

Sec 3.5 Limits at Infinity See anything?

Lesson 11.4 Limits at Infinity

26 – Limits and Continuity II – Day 2 No Calculator

3.5: ASYMPTOTES.

Objective: Section 3-7 Graphs of Rational Functions

Sec 4: Limits at Infinity

Limits at Infinity 3.5 On the agenda:

LIMITS AT INFINITY Section 3.5.

Limits at Infinity Section 3.5 AP Calc.

Asymptotes Horizontal Asymptotes Vertical Asymptotes

Warm Up – 12/4 - Wednesday Rationalize: − 5.

Section 8.4 – Graphing Rational Functions

Rational Functions A rational function f(x) is a function that can be written as where p(x) and q(x) are polynomial functions and q(x) 0 . A rational.

Graphing Rational Functions

Properties of Rational Functions

Properties of Rational Functions

Domain of Rational Functions

Asymptotes, End Behavior, and Infinite Limits

Presentation transcript:

limits at infinity (3.5) December 20th, 2011

I. limits at infinity Def. of Limit at Infinity: Let L be a real number. 1. The statement means that for each there exists an M>0 such that whenever x>M. 2. The statement means that for each there exists an N<0 such that whenever x<N.

II. horizontal asymptotes Def. of a Horizontal Asymptote: The line y=L is a horizontal asymptote of the graph of f if or.

*In other words, when a function f has a real number limit as or, it means that the function is approaching a horizontal asymptote at that limit value.

Thm. 3.10: Limits at Infinity: If r is a positive rational number and c is any real number, then. Furthermore, if is defined when x<0, then.

Ex. 1: Find.

Ex. 2: Find.

You Try: Find each limit. a. b. c.

Guidelines for Finding Limits at of Rational Functions: 1. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients. 3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.

Ex. 3: Find each limit. a. b.

You Try: Find each limit. a. b.

Ex. 4: Find.

III. infinite limits at infinity Def. of Infinite Limits at Infinity: Let f be a function defined on the interval. 1. means that for each positive number M, there is a corresponding number N>0 such that f(x)>M whenever x>N. 2. means that for each negative number M, there exists a corresponding number N>0 such that f(x) N. *This works for, too.

Ex. 5: Find.

Ex. 6: Find.

You Try: Find.