Presentation is loading. Please wait.

Presentation is loading. Please wait.

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.

Published byClarence Norman Modified over 9 years ago

Similar presentations

Presentation on theme: "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."— Presentation transcript:

1 1.3 – Continuity, End Behavior, and Limits

2

3

4 Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. a. f(x) = 3x – 2 if x > -3; at x = -3 2 – x if x < - 3

5 Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. a. f(x) = 3x – 2 if x > -3; at x = -3 2 – x if x < - 3 1. Find f(-3).

6 Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. a. f(x) = 3x – 2 if x > -3; at x = -3 2 – x if x < - 3 1. Find f(-3). f(-3) = 2 – (-3) = 5

7 Ex. 1 Determine whether each function is continuous at the given x value(s). Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. a. f(x) = 3x – 2 if x > -3; at x = -3 2 – x if x < - 3 1. Find f(-3). f(-3) = 2 – (-3) = 5, so f(-3) exists

8 2. Investigate values close to f(-3)

9 2. Investigate values close to f(-3) x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)5.15.015.001-10.997-10.97-10.7

10 2. Investigate values close to f(-3) As x -3 from left, f(x) 5 x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)5.15.015.001-10.997-10.97-10.7

11 2. Investigate values close to f(-3) As x -3 from left, f(x) 5 As x -3 from right, f(x) -11 x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)5.15.015.001-10.997-10.97-10.7

12 2. Investigate values close to f(-3) As x -3 from left, f(x) 5 As x -3 from right, f(x) -11 Since don’t approach same value, discontinuous and jump discontinuity. x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)5.15.015.001-10.997-10.97-10.7

13 b. f(x) = x + 3; at x = -3 and x = 3 x 2 – 9

14 b. f(x) = x + 3; at x = -3 and x = 3 x 2 – 9 1. Find f(-3) and f(3).

15 b. f(x) = x + 3; at x = -3 and x = 3 x 2 – 9 1. Find f(-3) and f(3). f(-3) = -3 + 3 = 0 = Ø (-3) 2 – 9 0

16 b. f(x) = x + 3; at x = -3 and x = 3 x 2 – 9 1. Find f(-3) and f(3). f(-3) = -3 + 3 = 0 = Ø (-3) 2 – 9 0 f(3) = 3 + 3 = 6 = Ø (3) 2 – 9 0

17 b. f(x) = x + 3; at x = -3 and x = 3 x 2 – 9 1. Find f(-3) and f(3). f(-3) = -3 + 3 = 0 = Ø (-3) 2 – 9 0 f(3) = 3 + 3 = 6 = Ø (3) 2 – 9 0 Since both f(-3) = Ø and f(3) = Ø, f(x) is discontinuous at both x = -3 and x = 3.

18 2. Investigate values close to f(-3) and f(3).

19 x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)-0.164-0.166-0.167 -0.169

20 2. Investigate values close to f(-3) and f(3). As x -3 from left, f(x) -0.167 x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)-0.164-0.166-0.167 -0.169

21 2. Investigate values close to f(-3) and f(3). As x -3 from left, f(x) -0.167 As x -3 from right, f(x) -0.167 x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)-0.164-0.166-0.167 -0.169

22 2. Investigate values close to f(-3) and f(3). As x -3 from left, f(x) -0.167 As x -3 from right, f(x) -0.167 Since they approach same value, limit exists. x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)-0.164-0.166-0.167 -0.169

23 2. Investigate values close to f(-3) and f(3). As x -3 from left, f(x) -0.167 As x -3 from right, f(x) -0.167 Since they approach same value, limit exists. x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)-0.164-0.166-0.167 -0.169 x2.92.992.9993.03.0013.013.1 f(x)f(x)-10-100-1000100010010

24 2. Investigate values close to f(-3) and f(3). As x -3 from left, f(x) -0.167 As x -3 from right, f(x) -0.167 Since they approach same value, limit exists. As x 3 from left, f(x) -∞ x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)-0.164-0.166-0.167 -0.169 x2.92.992.9993.03.0013.013.1 f(x)f(x)-10-100-1000100010010

25 2. Investigate values close to f(-3) and f(3). As x -3 from left, f(x) -0.167 As x -3 from right, f(x) -0.167 Since they approach same value, limit exists. As x 3 from left, f(x) -∞ As x 3 from right, f(x) ∞ x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)-0.164-0.166-0.167 -0.169 x2.92.992.9993.03.0013.013.1 f(x)f(x)-10-100-1000100010010

26 2. Investigate values close to f(-3) and f(3). As x -3 from left, f(x) -0.167 As x -3 from right, f(x) -0.167 Since they approach same value, limit exists. As x 3 from left, f(x) -∞ As x 3 from right, f(x) ∞ Since limit x -3 exists but f(-3) doesn’t, removable discontinuity. x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)-0.164-0.166-0.167 -0.169 x2.92.992.9993.03.0013.013.1 f(x)f(x)-10-100-1000100010010

27 2. Investigate values close to f(-3) and f(3). As x -3 from left, f(x) -0.167 As x -3 from right, f(x) -0.167 Since they approach same value, limit exists. As x 3 from left, f(x) -∞ As x 3 from right, f(x) ∞ Since limit x -3 exists but f(-3) doesn’t, removable discontinuity. Since limit x -3 doesn’t exist, infinite discontinuity. x-3.1-3.01-3.001-3-2.999-2.99-2.9 f(x)f(x)-0.164-0.166-0.167 -0.169 x2.92.992.9993.03.0013.013.1 f(x)f(x)-10-100-1000100010010

28 Ex. 2 Use the graph of the function to describe its end behavior.

29 Ex. 2 Use the graph of the function to describe its end behavior. lim f(x) = - ∞ x - ∞

30 Ex. 2 Use the graph of the function to describe its end behavior. lim f(x) = - ∞ x - ∞ lim f(x) = - ∞ x - ∞

Download ppt "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."

Similar presentations

3.5 Continuity & End Behavior

3.5 Continuity & End Behavior
What is a limit ? When does a limit exist? Continuity Discontinuity Types of discontinuity.

What is a limit ? When does a limit exist? Continuity Discontinuity Types of discontinuity.
Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2.

Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2.
6.1 – Graphing Systems of Equations

6.1 – Graphing Systems of Equations
Continuity and end behavior of functions

Continuity and end behavior of functions
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,

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.

 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.

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.
1.4 Continuity and One-Sided Limits This will test the “Limits” of your brain!

1.4 Continuity and One-Sided Limits This will test the “Limits” of your brain!
Splash Screen. Lesson Menu Five-Minute Check Then/Now New Vocabulary Key Concept:Limits Key Concept:Types of Discontinuity Concept Summary: Continuity.

Splash Screen. Lesson Menu Five-Minute Check Then/Now New Vocabulary Key Concept:Limits Key Concept:Types of Discontinuity Concept Summary: Continuity.
10.2: Infinite Limits. Infinite Limits When the limit of f(x) does not exist and f(x) goes to positive infinity or negative infinity, then we can call.

10.2: Infinite Limits. Infinite Limits When the limit of f(x) does not exist and f(x) goes to positive infinity or negative infinity, then we can call.
Continuity Section 2.3.

Continuity Section 2.3.
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.

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.
Warm up Evaluate the limit Section 2.4 Continuity SWBATSWBAT –Define continuity and its types.

Warm up Evaluate the limit Section 2.4 Continuity SWBATSWBAT –Define continuity and its types.
Limits Involving Infinity Section 2.2. ∞ Infinity Doesn’t represent a real number Describes the behavior of a function when the values in its domain or.

Limits Involving Infinity Section 2.2. ∞ Infinity Doesn’t represent a real number Describes the behavior of a function when the values in its domain or.
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 =

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

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

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.
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.

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.

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.

Similar presentations

Ads by Google