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1.5: Limits Involving Infinity Learning Goals: © 2009 Mark Pickering Calculate limits as Identify vertical and horizontal asymptotes
Example Find the limit if it exists: In the previous lesson, we had this problem:
Example Find the limit if it exists: How does this problem differ from the previous problem?
Important Idea The above symbols describe the increasing or decreasing of a value without bound. Infinity is not a number.
Example Use the table feature of your calculator to estimate: if it exists.
Definition if they exist, means y = b is a horizontal asymptote b or
Try This Use the graph and table features of your calculator to find any horizontal asymptotes for:
Try This Use the graph and table features of your calculator to find any horizontal asymptotes for: &
Important Idea Functions involving radicals may have 2 horizontal asymptotes Horizontal asymptotes are always written as
Example Find the limit, if it exists: Indeter- minate form Divide top & bottom by highest power of x in denominator.
Try This Find the limit, if it exists: DNE or
Try This Find the limit, if it exists: 0
Try This Find the limit, if it exists:
Analysis In the last 3 examples, do you see a pattern? The highest power term is most influential.
Important Idea If the degree of the top is greater than the degree of the bottom, For any rational function
Important Idea If the degree of the bottom is greater than the degree of the top, For any rational function
Important Idea If the degree of the top is the same as the degree of the top, the limit as For any rational function is the ratio of the leading coefficients.
Try This Find the limit if it exists:
Example Functions may approach different asymptotes as and as Consider each limit separately…
Example As eventually x >0. Divide radical by and divide non-radical by x. Find the limit:
Example As eventually x <0. Divide radical by and divide non-radical by x. Find the limit:
Try This Using algebraic techniques, find the limit if it exists. Confirm your answer with your calculator.
Analysis The sine function oscillates between +1 and -1 1
Analysis 1 The limit does not exist due to oscillation.
Analysis and Therefore, by the Sandwich Theorem,
A function has an infinite limit at a if f(a) as x a. f(x) is unbounded at x=a. f(x) Definition
A function has an infinite limit at a if f(a) as x a. f(x) is unbounded at x=a. f(x) Definition The line x = a is a vertical asymptote
Important Idea A vertical asymptote is written x = a A horizontal asymptote is written y = a where a is any real number.
Try This What is the limit as x 1 from the left and from the right? x =1 What is the vertical asymptote?
Solution x =1 Vertical asymptote: x =1
Try This What is the limit as x 1 from the left and from the right? x =1 What is the vert. asymptote?
Solution x =1 Vertical asymptote: x =1
Definition The value(s) that make the denominator of a rational function zero is a vertical asymptote.
Example Determine all vertical asymptotes of Steps: 1. Factor & cancel if possible 2. Set denominator to 0 & solve
Example Find the limit if it exists: 1. When you substitute x =1, do you get a number /0 or 0/0? The questions… 2. What is happening at x = a value larger than 1?
Try This Find the limit if it exists: +
Find all vertical asymptotes for for Hint: since, Example where does ?
Lesson Close Limits are the foundation of both differential and integral Calculus. We will develop these ideas in chapter 3.
Practice 76/1-7,13,15,27-33,59 (all odd)
3-5: Limits at Infinity Objectives: ©2003 Roy L. Gover Discuss the end behavior of functions on an infinite interval. Understand horizontal.
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:
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 Objectives: Students will be able to evaluate limits as Students will be able to find horizontal and vertical asymptotes.
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:
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:
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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.
Infinite Limits and Limits to Infinity: Horizontal and Vertical Asymptotes.
0-3: Rational Functions & Asymptotes Objectives: Determine horizontal, vertical & slant asymptotes Graph rational functions ©2002 Roy L. Gover
Aim: How do find the limit associated with horizontal asymptote? Do Now: 1.Sketch f(x) 2.write the equation of the vertical asymptotes.
Rational Functions and Their Graphs Section 2.6 Page 326.
11.2: Graphing Rational Functions Algebra 1: May 1, 2015.
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 at Infinity: End behavior of a Function. Limits of Polynomials as x approaches infinity or negative infinity The term with the highest power will.
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.
Table of Contents Rational Functions: Horizontal Asymptotes Horizontal Asymptotes: A horizontal asymptote of a rational function is a horizontal line (equation:
Review: Graph: #3 on Graphing Calc to see how it looks. › HA, VA, Zeros, Y-int.
L IMITS AND L IMITS AT INFINITY Limit Review 1. Limits can be calculated 3 ways Numerically Graphically Analytically (direct substitution) Properties.
Warm Up - Factor the following completely : 1. 3x 2 -8x x x x 3 +2x 2 -4x x 2 -x x (3x-2)(x-2) 11(x+3)(x-3)
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Lines that a function approaches but does NOT actually touch.
Ch. 9.3 Rational Functions and Their Graphs. For each rational function, find any points of discontinuity. ALGEBRA 2 LESSON 9-3 Rational Functions and.
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Objectives: To evaluate limits numerically, graphically, and analytically. To evaluate infinite limits.
Chapter 3 Limits and the Derivative Section 2 Infinite Limits and Limits at Infinity.
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H.Melikian1 §10.2 Infinite Limits and Limits at Infinity Dr.Hayk Melikyan Departmen of Mathematics and CS The student will be able to.
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.
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag Example: ,000,000 Example:
–1 –5–4–3–2– Describe the continuity of the graph. Warm UP:
1 Find the domains of rational functions. Find the vertical and horizontal asymptotes of graphs of rational functions. 2.6 What You Should Learn.
1. Steps to finding a limit 2. Limits approaching a value 3. One-Sided limits 4. Limits at infinity 5. Limits with radicals 6. Trig Limits 7. Limits of.
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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.
Table of Contents Rational Functions: Slant Asymptotes Slant Asymptotes: A Slant asymptote of a rational function is a slant line (equation: y = mx + b)
Rational Functions 4-2 A Rational Function is an equation in the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions, and q(x) does.
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.
Class Work 1.Find the real zeros by factoring. P(x) = x 4 – 2x 3 – 8x Divide. 3.Find all the zeros of the polynomial. P(x) = x 3 – 2x 2 + 2x – 1.
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OBJECTIVE: 1. DEFINE LIMITS INVOLVING INFINITY. 2. USE PROPERTIES OF LIMITS INVOLVING INFINITY. 3. USE THE LIMIT THEOREM. 14.5Limits Involving Infinity.
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