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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 Determine limits at infinity
Chapter 3: Applications of Differentiation L3.5 Limits at Infinity.
Horizontal and Vertical Asymptotes. Vertical Asymptote A term which results in zero in the denominator causes a vertical asymptote when the function is.
Rational Expressions, Vertical Asymptotes, and Holes.
Ch. 9.3 Rational Functions and Their Graphs
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag Example: ,000,000 Example:
A rational function is a function of the form: where p and q are polynomials.
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)
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.
Table of Contents Rational Functions: Horizontal Asymptotes Horizontal Asymptotes: A horizontal asymptote of a rational function is a horizontal line (equation:
Table of Contents Rational Functions: Slant Asymptotes Slant Asymptotes: A Slant asymptote of a rational function is a slant line (equation: y = mx + b)
Chapter 3 Limits and the Derivative
2.7 – Graphs of Rational Functions. By then end of today you will learn about……. Rational Functions Transformations of the Reciprocal function Limits.
ACT Class Openers:
1 Find the domains of rational functions. Find the vertical and horizontal asymptotes of graphs of rational functions. 2.6 What You Should Learn.
Infinite number of rooms Hotel Paradox Proofs 1/3= inf = inf implies inf = 0 Intro to Calculus Rainbow Bridge, find the area under the curve.
Rational Functions 4-2.
Infinite Limits and Limits to Infinity: Horizontal and Vertical Asymptotes.
OBJECTIVE: 1. DEFINE LIMITS INVOLVING INFINITY. 2. USE PROPERTIES OF LIMITS INVOLVING INFINITY. 3. USE THE LIMIT THEOREM. 14.5Limits Involving Infinity.
Objectives: To evaluate limits numerically, graphically, and analytically. To evaluate infinite limits.
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