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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)
2.2 Limits Involving Infinity Finite Limits as – The symbol for infinity does not represent a real number. – We use infinity to describe the behavior of.
Chapter 3: Applications of Differentiation L3.5 Limits at Infinity.
Find the zeros of the following function F(x) = x 2 -1 Factor the following function X 2 + x – 2 Simplify the expression: X 2 + x – 2 x 2 – x – 6.
LIMITS OF FUNCTIONS. INFINITE LIMITS; VERTICAL AND HORIZONTAL ASYMPTOTES; SQUEEZE THEOREM OBJECTIVES: define infinite limits; illustrate the infinite.
ASYMPTOTES TUTORIAL Horizontal Vertical Slant and Holes.
Linear & Quadratic Functions PPT What is a function? In order for a relation to be a function, for every input value, there can only be one output.
Growth and Decay Exponential Models. Differs from polynomial functions. Polynomial Functions have exponents of whole numbers Exponential Functions have.
Rational Functions. Definition: A Rational Function is a function in the form: f(x) = where p(x) and q(x) are polynomial functions and q(x) 0. In this.
INFINITE LIMITS. LIMITS What is Calculus? What are Limits? Evaluating Limits –Graphically –Numerically –Analytically What is Continuity? Infinite Limits.
Copyright © Cengage Learning. All rights reserved. 2.7 Graphs of Rational Functions.
Vertical Asymptotes (1) Vertical asymptotes exist when the rational function is in lowest terms and its denominator polynomial can be equal to 0 for some.
I n f i n i t e L i m i t s a t L i m i t s & I n f i n i t y I n f i n i t e L i m i t s a t L i m i t s & I n f i n i t y I n f i n i t e L i m i t s.
RATIONAL FUNCTIONS A rational function is a function of the form: where p and q are polynomials.
9.3 Rational Functions and Their Graphs. If the graph is not continuous at x = a then the function has a point of discontinuity at x = a.
2.2 Limits Involving Infinity Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.
LIMITS What is Calculus? What are Limits? Evaluating Limits –Graphically –Numerically –Analytically What is Continuity? Infinite Limits This presentation.
Page 0 Introduction to Limits. Page 1 Definition of the limit of f(x) as x approaches a: We write and say the limit of f(x), as x approaches a, equals.
Page 20 Why [0/0] and [ – ] Are Indeterminate Forms Whereas k/0 Isnt? [0/0] is an indeterminate form because a limit of this form is unpredictable. If.
AP Calculus Review First Semester Differentiation to the Edges of Integration Sections , 3.9, (7.7)
Copyright © Cengage Learning. All rights reserved. 2.6 Rational Functions and Asymptotes.
3.4 Rational Functions I. A rational function is a function of the form Where p and q are polynomial functions and q is not the zero polynomial. The domain.
Page 6 As we can see, the formula is really the same as the formula. So, Furthermore, if an equation of the tangent line at (a, f(a)) can be written as:
3.7 Graphs of Rational Functions. A rational function is a quotient of two polynomial functions. A rational function is a quotient of two polynomial functions.
Weve looked at linear and quadratic functions, polynomial functions and rational functions. We are now going to study a new function called exponential.
1 Graph Sketching: Asymptotes and Rational Functions OBJECTIVES Find limits involving infinity. Determine the asymptotes of a function’s graph.
Page 26 Limit and Continuity of Piecewise-defined Functions x y h/d When we write f(x) = |2x + 5| as f(x) =, the new format is called a piecewise- defined.
Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Objectives Identify and evaluate rational functions. Graph a rational function, find its.
Polynomial Functions and Their Graphs. Definition of a Polynomial Function Let n be a nonnegative integer and let a n, a n- 1,…, a 2, a 1, a 0, be real.
1 6.3 Exponential Functions In this section, we will study the following topics: Evaluating exponential functions with base a Graphing exponential functions.
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