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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 values of x where f (x) is undefined (sort of). and are examples of graphs that have a hole.
Ex. Find all asymptotes of, then sketch the graph.
means that x approaches 2 from the right (larger than 2) means that x approaches 2 from the left (smaller than 2) One-Sided Limits Ex.
Thm. The limit exists if both sides agree.
Ex. For the function given, find: a. b. c.
Ex. Find if
Def. (loose) A function is continuous on an interval if the graph has no gaps, jumps, or breaks on the interval. Ex. Is continuous on [0,5]?
Def. (tight) A function f (x) is continuous on an interval if, for all points c on the interval: i. exists ii. exists iii.
Ex. Let Find a value of B so that f (x) is continuous at x = 0.
The test on Chapter 1 will be on Monday. Next class we will review and Ill pass out a Sample Test so you know what types of questions I can ask.
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.
Section Continuity. continuous pt. discontinuity at x = 0 inf. discontinuity at x = 1 pt. discontinuity at x = 3 inf. discontinuity at x = -3 continuous.
Section Continuity 2.2. A function f(x) is continuous at x = c if and only if all three of the following tests hold: f(x) is right continuous at.
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.
The derivative and the tangent line problem (2.1) October 8th, 2012.
Writing Equations of Lines. Find the equation of a line that passes through (2, -1) and (-4, 5).
Unit 7 –Rational Functions Graphing Rational Functions.
Graph of Exponential Functions Chapter 3 Lesson G.
Lesson 2.6 Rational Functions and Asymptotes. Graph the function: Domain: Range: Increasing/Decreasing: Line that creates a split in the graph:
Chapter 7 Polynomial and Rational Functions with Applications Section 7.2.
Objective: Sketch the graphs of tangent and cotangent functions.
Limits at infinity (3.5) December 20th, I. limits at infinity Def. of Limit at Infinity: Let L be a real number. 1. The statement means that for.
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.
Infinite Limits 1.5. An infinite limit is a limit in which f(x) increases or decreases without bound as x approaches c. Be careful…the limit does NOT.
Definition of a Rational Function A rational function is a quotient of polynomials that has the form The domain of a rational function consists of all.
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.
LESSON 20 PREREQ B: Writing the Equation of a Line.
Section 5.3 – Limits Involving Infinity. X X Which of the following is true about I. f is continuous at x = 1 II. The graph of f has a vertical asymptote.
2-2 LIMITS INVOLVING INFINITY. Let’s start by considering and y = 0 is a horizontal asymptote because Def: The line y = b is a horizontal asymptote of.
LESSON 5 Section 6.3 Trig Functions of Real Numbers.
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.
Q2-1.1a Graphing Data on the coordinate plane. Graph each number on a number line a) 4 b) -3 c) 2.7 d) -1.2.
Extrema on an interval (3.1) November 15th, 2012.
Objectives: 1.Be able to define continuity by determining if a graph is continuous. 2.Be able to identify and find the different types of discontinuities.
Concavity & the second derivative test (3.4) December 4th, 2012.
Graphing Rational Functions. I. Rational Functions Let P(x) and Q(x) be polynomial functions with no common factors and, then is a rational function.
Lesson 2.7 Page 161: #1-33 (EOO), 43, 47, 59, & 63 EXTRA CREDIT: Pages (Do any 20 problems…You choose ) STUDY: Chapter 2 Exam 10/15 (Make “CHEAT.
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?
Graphing Rational Functions Objective: To graph rational functions without a calculator.
Linear Models & Rates of Change (Precalculus Review 2) September 9th, 2015.
Continuity Lesson Learning Objectives Given a function, determine if it is continuous at a certain point using the three criteria for continuity.
Rational Functions Intro - Chapter 4.4. Let x = ___ to find y – intercepts A rational function is the _______ of two polynomials RATIO Graphs of Rational.
Increasing & Decreasing Functions & The First Derivative Test (3.3) November 29th, 2012.
Infinite Limits Determine infinite limits from the left and from the right. Find and sketch the vertical asymptotes of the graph of a function.
Limits and Their Properties. Limits We would like to the find the slope of the tangent line to a curve… We can’t because you need TWO points to find a.
Lesson 2.6 Read: Pages Page 152: #1-37 (EOO), 47, 49, 51.
Riemann sums & definite integrals (4.3) January 28th, 2015.
A function, f, is continuous at a number, a, if 1) f(a) is defined 2) exists 3)
Lesson 5-1. The ___________ of a line is a number determined by any two points on the line. It is the ratio of the ___________ (vertical change) over.
8-3 Rational Functions Unit Objectives: Graph a rational function Simplify rational expressions. Solve a rational functions Apply rational functions to.
Limit & Derivative Problems Problem…Answer and Work…
2.5 RATIONAL FUNCTIONS DAY 2 Learning Goals – Graphing a rational function with common factors.
Rational Functions Objective: Finding the domain of a rational function and finding asymptotes.
Implicit differentiation (2.5) October 29th, 2012.
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.
ACT Class Opener: om/coord_1213_f016.htm om/coord_1213_f016.htm
9.3 – Rational Function and Their Graphs. Review: STEPS for GRAPHING HOLES ___________________________________________ EX _________________________________________.
Slopes of Lines Chapter 3-3. Finding Slope of a Line.
Section c Let h be a function defined for all such that h(4) = -3 and the derivative of h is given by for. Write an equation for the line tangent.
Notes Over 9.2 Graphing a Rational Function The graph of a has the following characteristics. Horizontal asymptotes: center: Then plot 2 points to the.
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