Continuity of Function at a Number

Slides:

Advertisements

Similar presentations

INFINITE LIMITS.

Advertisements

LIMITS What is Calculus? What are Limits? Evaluating Limits

1.2 Functions & their properties

Graphing Rational Functions

What is a limit ? When does a limit exist? Continuity Discontinuity Types of discontinuity.

ACT Class Opener: om/coord_1213_f016.htm om/coord_1213_f016.htm

Warm-Up/Activator Sketch a graph you would describe as continuous.

Continuity When Will It End. For functions that are "normal" enough, we know immediately whether or not they are continuous at a given point. Nevertheless,

 A continuous function has no breaks, holes, or gaps  You can trace a continuous function without lifting your pencil.

Domain & Range Domain (D): is all the x values Range (R): is all the y values Must write D and R in interval notation To find domain algebraically set.

3.7 Graphing Rational Functions Obj: graph rational functions with asymptotes and holes and evaluate limits of rational functions.

Continuity 2.4.

Section Continuity. continuous pt. discontinuity at x = 0 inf. discontinuity at x = 1 pt. discontinuity at x = 3 inf. discontinuity at x = -3 continuous.

Continuity TS: Making decisions after reflection and review.

Continuity and Discontinuity

Section 1.4 Continuity and One-sided Limits. Continuity – a function, f(x), is continuous at x = c only if the following 3 conditions are met: 1. is defined.

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.

Limits. a limit is the value that a function or sequence "approaches" as the input approaches some value.

Asymptotes Objective: -Be able to find vertical and horizontal asymptotes.

Lesson 2.6 Rational Functions and Asymptotes. Graph the function: Domain: Range: Increasing/Decreasing: Line that creates a split in the graph:

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.

What is the symmetry? f(x)= x 3 –x.

2.2 Limits Involving Infinity Goals: Use a table to find limits to infinity, use the sandwich theorem, use graphs to determine limits to infinity, find.

Continuity of A Function. 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 x =

2.5 – Continuity A continuous function is one that can be plotted without the plot being broken. Is the graph of f(x) a continuous function on the interval.

A function, f, is continuous at a number, a, if 1) f(a) is defined 2) exists 3)

2.3 Continuity.

Continuity of A Function 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.

1.4 One-Sided Limits and Continuity. Definition A function is continuous at c if the following three conditions are met 2. Limit of f(x) exists 1. f(c)

Informal Description f(x) is continuous at x=c if and only if there are no holes, jumps, skips or gaps in the graph of f(x) at c.

Review Limits When you see the words… This is what you think of doing…  f is continuous at x = a  Test each of the following 1.

Removable Discontinuities & Vertical Asymptotes

I can graph a rational function.

Lesson 8-3: Graphing Rational Functions

Section Continuity 2.2.

LIMITS OF FUNCTIONS. CONTINUITY Definition (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.

1.3 – Continuity, End Behavior, and Limits. Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity.

2.4 Continuity Objective: Given a graph or equation, examine the continuity of a function, including left-side and right-side continuity. Then use laws.

1.3 Limits, Continuity, & End Behavior September 21 st, 2015 SWBAT estimate a limit of a graphed function. SWBAT identity a point of discontinuity utilizing.

1.5 Infinite Limits. Find the limit as x approaches 2 from the left and right.

Point Value : 20 Time limit : 1 min #1. Point Value : 20 Time limit : 1 min #2.

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.

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

What is a limit ? When does a limit exist? Continuity Continuity of basic algebraic functions Algebra of limits The case 0/0.

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.

Asymptotes of Rational Functions 1/21/2016. Vocab Continuous graph – a graph that has no breaks, jumps, or holes Discontinuous graph – a graph that contains.

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.

Continuity and One-Sided Limits

Decide whether each of the following is continuous or not.

4.4 Rational Functions A Rational Function is a function whose rule is the quotient of two polynomials. i.e. f(x) = 1

Ch. 2 – Limits and Continuity

Limits, Asymptotes, and Continuity

3.7 Graphs of Rational Functions

Ch. 2 – Limits and Continuity

Continuity and One Sided Limits

26 – Limits and Continuity II – Day 2 No Calculator

5-Minute Check Lesson 3-5 Find the inverse of y = - 3x + 4 Then graph both on the same coordinate axes. What is the relationship between a graph of a.

Continuous & Types of Discontinuity

MATH(O) Limits and Continuity.

Limits involving infinity

Notes Over 9.3 Graphing a Rational Function (m < n)

Section 2.2 Objective: To understand the meaning of continuous functions. Determine whether a function is continuous or discontinuous. To identify the.

26 – Limits and Continuity II – Day 1 No Calculator

27 – Graphing Rational Functions No Calculator

EQ: What other functions can be made from

Which is not an asymptote of the function

Presentation transcript:

Continuity of Function at a Number A function y = f(x) is continuous at a number a if and only if the following are satisfied: If one or more of these three conditions are not satisfied, then the function f(x) is said to be discontinuous at a.

equal f is continuous at x = 1. Graphically, it means there is no break in the graph of f(x) at x = 1.

h is not continuous at x = 1. Not equal h is not continuous at x = 1. Graphically, it means there is a break ( jump) in the graph of h(x) at x = 1.

f is not continuous at x = 2. Graphically, it means there is a break (hole) in the graph of f(x) at x = 2.

h is not continuous at x = -2. Graphically, it means there is a break in the graph of h(x) at x = -2 (vertical asymptote).

x = -2 (vertical asymptote) y = 2 is a horizontal asymptote - 4 - 3 - 1 y = h(x) 7 12 - 8 (0, -3) , (3, 0) vertical asymptote x = -2 y = 2 is a horizontal asymptote 9

Types of Discontinuity Removable discontinuity – discontinuity where Graphically, it means there is a break (hole) in the graph of f(x) at x = a and this hole can be patched by redefining f(x) to be equal to its limit L when x = a. Redefine

f has removable discontinuity at x = 2. Redefine

Types of Discontinuity b) Essential discontinuity – discontinuity where (i) Jump discontinuity – essential discontinuity where Graphically, it means there is a break (jump) in the graph of f(x) at x = a. The graph approaches two different points as x approaches a from the left and right.

Not equal h has a jump essential discontinuity at x = 1.

Types of Discontinuity b) Essential discontinuity – discontinuity where (ii) Infinite discontinuity – essential discontinuity where Graphically, it means there is a break in the graph of f(x) at x = a where x = a is a vertical asymptote.

h has an infinite essential discontinuity at x = -2.

x = -2 (vertical asymptote) y = 2 is a horizontal asymptote - 4 - 3 - 1 y = h(x) 7 12 - 8 (0, -3) , (3, 0) vertical asymptote x = -2 y = 2 is a horizontal asymptote 19