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Title: 1.4 **Continuity** **and** **one**-**sided** **limits** Author: SWSD Last modified by: SWSD Created Date: 2/8/2007 4:24:43 PM Document presentation format: On-screen Show

Times New Roman Default Design MathType 5.0 Equation **Limits** **and** **Continuity** Limit PowerPoint Presentation PowerPoint Presentation PowerPoint Presentation PowerPoint Presentation The Definition of Limit PowerPoint Presentation Examples PowerPoint Presentation **One**-**Sided** **Limits** PowerPoint ...

**Continuity** **and** **One**-**Sided** **Limits** Lesson 2.4 * * * * Intuitive Look at **Continuity** A function without breaks or jumps The graph can be drawn without lifting the pencil * **Continuity** at a Point A function can be discontinuous at a point A hole in the function **and** the function not defined at that ...

-Students will determine **one**-**sided** **limits** **and** **continuity** on a closed interval-Students will use properties of **continuity**-Students will understand **and** use Intermediate Value Theorem **Continuity** at a point x = c If a function is continuous at x = c, this means there are no interruptions ...

Title: 1-4: **Continuity** **and** **One**-**Sided** **Limits** Author: Roy Gover Last modified by: Michael Efram Created Date: 9/13/2002 10:21:30 PM Document presentation format

... because the **one**-**sided** **limits** match the value of the function 1 2 3 4 1 2 f is continuous at point c if: 1) f is defined there. 2) exists 3 ) Continuous ... [-1,1]. Ex. Discuss the **continuity** The two functions **and** are continuous over all reals. The only point in question is at x ...

(Use your three criteria.) **One**-**sided** **Limits** We’ve learned: In this lesson, you will get problems that ask you to evaluate **one**-**sided** **limits**. Example 2 Evaluate: ... determine if it is continuous at a certain point using the three criteria for **continuity**. Evaluate a **one**-**sided** limit.

1.4 **Continuity** **and** **One**-**Sided** **Limits** This will test the “**Limits**” of your brain! Definition of **Continuity** A function is called continuous at c if the following three conditions are met: 1.

**One**-**sided** **Continuity** **Continuity** on Intervals Section 2.5 The Pinching Theorem; Trigonometric **Limits** ... = 0 lim x2 – 9 = 0. x → 3 x → 3 h → 3 **One**-**sided** **limits** give us a simple way of determining whether or not a (two-**sided**) limit exists: Example 9 For the function defined by ...

... because the **one**-**sided** **limits** match the value of the function 1 2 3 4 1 2 jump infinite oscillating Essential Discontinuities: Removable Discontinuities: (You can fill the hole.) Removing a ... **Continuity** Author: Gregory Kelly Last modified by: Greg & Vickie Kelly Created Date: 3/10/2003 8:30 ...

§ 1-4 **Limits** **and** **Continuity** The student will learn about: **limits**, **limits**, finding **limits**, **one**-**sided** **limits**, infinite **limits**, **and** **continuity**. * **Limits** The word “limit” is used in everyday conversation to describe the ultimate behavior of something, as in the “limit of **one**’s endurance ...

... **One** **Sided** **Limits** **and** **Continuity** A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a, b) **and** The function is continuous from the right at a **and** continuous from the left at b. f(x) is continuous on [-3,2] ...

... because the **one**-**sided** **limits** match the value of the function 1 2 3 4 1 2 2.3 **Continuity** Definition: **Continuity** A function is continuous at a number a if That is, 1. f(a) is defined 2. exists 3. 2.3 **Continuity** Definition: **One** **Sided** **Continuity** A function f is continuous from the right at a ...

Limit **and** **Continuity** 2.1 Rate of Change **and** **Limits** (1) Average **and** Instantaneous Speed 2.1 Rate of ... Properties of Limit 2.1 Rate of Change **and** **Limits** (14) **One**-**sided** **and** Two-**sided** **Limits** 2.1 Rate of Change **and** **Limits** (15, Example 7) **One**-**sided** **and** Two-**sided** **Limits** 2.1 Rate of Change ...

Notice this difference quotient is the change in y over the change in x. calculate **one** **sided** **limits** 7 5 Since the **one**-**sided** **limits** are not equal, ... The function in (a) is continuous at x = 0; the functions in (b) through ( f ) are not. 2.5 **Continuity** in (a) at x = 0 but not in other graphs.

**One**-**Sided** **and** Two-**Sided** **Limits** Let’s examine the following graph to further explore right **and** left hand **limits**. o 1 ... **Limits** **and** **Continuity** 2.1 Rates of Change **and** **Limits** Slide 3 Slide 4 Slide 5 Slide 6 Slide 7 Slide 8 Slide 9 Slide 10 Slide 11 Slide 12 Slide 13 Slide 14 Slide ...

**Continuity** **and** Discontinuity Hien Pham **and** Kim Lai ... only the **one**-**sided** **limits** need to equal the function Cusp A cusp is a point on the graph at which the function is continuous but the derivative is discontinuous Verbally: ...

... **Limits** - **Continuity** from Paul Dawkins A great discussion plus graphs from Stefan Waner at Hofstra U **Continuity** **and** Differentiability then do the **Continuity** ... Online Examples Follow this link to **One**-**sided** derivatives from IES Software **And** then follow this link to Investigating ...

The Breaking of the Three Rules of **Continuity** Hole Jump Gap jump infinite oscillating Non-removable Discontinuities: Removable Discontinuities: ... Explain Set up **and** solve two **one** – **sided** **limits**. What kind of discontinuity occurs at x = 2? At x = 1 Theorem for Thought…Not for Copying!

... does not exist at x=c. http://www.algebra-class.com/image-files/step-functions-5.gif **One**-**Sided** **Limits** Only ... a Point on an Open Interval **Continuity** at a Point on an Open Interval **One**-**Sided** **Limits** **One**-**Sided** **Limits** **One** **Sided** **Limits** Existence of a Limit **Continuity** at a Point ...

Mika Seppälä: **Limits** **and** **Continuity** **One**-**sided** **Limits** Problem 10 Solution Mika Seppälä: **Limits** **and** **Continuity** **Continuity** Problem 11 Solution Mika Seppälä: **Limits** **and** **Continuity** **Continuity** Problem 12 Solution Mika Seppälä: ...

... **Limits** by Rewriting **Limits** by Rewriting **Limits** by Rewriting **Limits** by Rewriting **Limits** by Rewriting **Limits** by Rewriting **One**-**sided** **Limits** **Continuity** **Continuity** **Continuity** **Continuity** **Continuity** Solved Problems on **Limits** **and** **Continuity** Overview of Problems Overview of Problems Main ...

... **Limits** by Rewriting **Limits** by Rewriting **Limits** by Rewriting **Limits** by Rewriting **Limits** by Rewriting **Limits** by Rewriting **One**-**sided** **Limits** **Continuity** **Continuity** **Continuity** **Continuity** **Continuity** Solved Problems on **Limits** **and** **Continuity** Overview of Problems Overview of Problems Main ...

3.5 - Absolute **Limits** **and** **Continuity** Absolute **Limits**: ... To determine the limit we must use "**ONE**-**SIDED** **LIMITS**". Ex.) Determine the limit of each of the following functions. a) b) **Continuity**: * The graph of a continuous function exits without a break.

2.1 Rates of Change **and** **Limits** Quick Review Quick Review Quick Review What you’ll learn about Average **and** Instantaneous Speed Definition of Limit Properties of **Limits** **One**-**Sided** **and** Two-**Sided** **Limits** Sandwich Theorem Essential Question How can **limits** be used to describe **continuity**, the ...

Chapter 2 **Limits** **and** **Continuity** 2.1 Rates of Change **and** **Limits** Quick Review Quick Review Quick Review Solutions Quick Review Solutions What you’ll learn about Average **and** Instantaneous Speed Definition of Limit Properties of **Limits** **One**-**Sided** **and** Two-**Sided** **Limits** Sandwich Theorem …**and** why ...

... No Limit **One**-**sided** **Limits** (1) **One**-**sided** **Limits** (2) **One**-**sided** **Limits** (3) Continuous Functions Symmetric Functions (1) Symmetric Functions (2) ... **Limits** of Functions **and** **Continuity** Subject: Calculus Author: Mika Seppälä Keywords: functions, **limits**, **continuity**, injective, surjective

... because the **one**-**sided** **limits** match the value of the function 1 2 3 4 1 2 jump infinite oscillating Essential Discontinuities: Removable Discontinuities: (You can fill the hole ... **Continuity** Author: Gregory Kelly Last modified by: greenbay Created Date: 3/10/2003 8:30:45 PM Document ...

... [0, 1] such that cont’d * **Continuity** **and** **One**-**Sided** **Limits** 1.4 **Continuity** at a Point **and** on an Open Interval Three values of x at which the graph of f is not continuous. At all other points in the interval (a, b), ...

... **Limits** **and** **Continuity** (Two **Sided**) **Limits** **One** **Sided** **Limits** (**One** **Sided**) Right-Hand **Limits** (**One** **Sided**) Left-Hand **Limits** Examples … When Left & Right **Limits** are Equal A special & important limit. A special & important limit.

These **one**-**sided** **limits** are called the derivatives from the left **and** from the right, ... Differentiability **and** **Continuity** You can verify this by observing that **and** Although it is true that differentiability implies **continuity** the converse is not true.

Functions, **Limits**, **and** the Derivative 2 Functions **and** Their Graphs The Algebra of Functions Functions **and** Mathematical Models **Limits** **One**-**Sided** **Limits** **and** **Continuity**

Determine **one**-**sided** **limits** **and** **continuity** on a closed interval. Use properties of **continuity**. Understand **and** use the Intermediate Value Theorem. **Continuity** at a ...

Top ten famous **limits**: 6-10 Basic properties of **limits** II. Two-**sided** **and** **one**-**sided** **limits**: IV. Squeeze theorem: Let’s work ... Squeeze theorem: Let’s work through a few: Now you try this **one**... **Continuity** Intermediate value theorem Maple graph We had f(1.75) < 0 **and** f(2) > 0. So the ...

To study **one**-**sided** **limits**, infinite **limits**, **and** **limits** at infinity. ... Solution: Chapter 10: **Limits** **and** **Continuity** 10.1 **Limits** Chapter 10: **Limits** **and** **Continuity** 10.1 **Limits** Example 3 – Applying Limit Properties 1 **and** 2 Chapter 10: ...

Chapter 2 **Limits** **and** **Continuity** 2.1 Rates of Change **and** **Limits** 2.2 Calculating **Limits** Using the **Limits** Laws 2.3 The Precise Definition of a Limit 2.4 **One**-**Sided** **Limits** **and** **Limits** at Infinity 2.5 Infinite **Limits** **and** Vertical Asymptotes 2.6 **Continuity** 2.7 Tangents **and** Derivatives Chapter 2 ...

Limit Definition of **Continuity** Limit Definition of **Continuity** Most of the techniques of calculus require that ... because the **one**-**sided** **limits** match the value of the function 1 2 3 4 1 2 jump infinite oscillating Essential Discontinuities: Removable Discontinuities: (You can fill the ...

Figure 1.31 **One**-**Sided** **Limits** **and** **Continuity** on a Closed Interval Example 4 – **Continuity** on a Closed Interval Discuss the **continuity** of f(x) = Solution: The domain of f is the closed interval [–1, 1]. At all points in the open interval ...

This requires that the **one**-**sided** **limits** exist **and** are equal I. Differentiability **and** **Continuity** x f(x) -.5 -1/-.5 -.1 -1/-.1 -.01 -1/-.01 .5 ... So, **continuity** does not imply differentiability (Sharp turns in graphs **and** vertical tangents). I. Differentiability **and** **Continuity** II ...

For instance, if n is an even integer, **One**-**Sided** **Limits** **and** **Continuity** on a Closed Interval Example 2 – A **One**-**Sided** Limit Find the limit of f(x) = as x approaches –2 from the right. Solution: As shown in Figure 1.29, the limit as x ...

Example: 11.3 **Limits** & **Continuity** Objectives Determine whether a function is continuous at a number Determine for what numbers a function is discontinuous A ... therefore a=-2. The function values approached 12, so L=12. **One**-**sided** **limits** Sometimes we’re interested in what happens as values ...

These **one**-**sided** **limits** are called the derivatives from the left **and** from the right, ... Differentiability Implies **Continuity** If f is differentiable at then f is continuous at . Title: POWER SERIES Author: MSE - Shannon Gracey Last modified by: LENOVO USER Created Date: 4/14/2008 5:56:00 PM

... 4 Proving **Limits** Proving **Limits** **One** **Sided** **Limits** **Limits** as x ± **Limits** as x ± When a limit DNE **Continuity** Common Points of Discontinuity Continuous Functions Intermediate Value Theorem Derivatives MTH 251 ...

... Problem Slope at Specific Point vs. Formula Definition of the Derivative Finding a Derivative Differentiability **and** **Continuity** Differentiability **and** ... graph at more than **one** point. 1 2 This is an ... at 2? m = 1 m = -1 YES **One**-**sided** **limits** are not equal, f ...

**One**-**Sided** **Limits** Similarly, if we require that x be greater than a, ... Solution Infinite **Limits** 投影片 36 **Continuity** **Continuity** **Continuity** Discontinuity **Continuity** **Continuity** **Continuity** **Continuity** **Continuity** **Continuity** Intermediate Value Theorem 投影片 48 ...

Differentiability **and** **Continuity** Note that the existence of the limit in this alternative form requires that the **one**-**sided** **limits** exist **and** ... you can extend the definition of a vertical tangent line to include the endpoints by considering **continuity** **and** **limits** from the right (for x ...

Since e is arbitrary, |K-L|<2e implies K=L. e-d definition of **one**-**sided** **limits** ... Proof of uniqueness of **limits** e-d definition of **one**-**sided** **limits** Useful notations M-d definition of infinite **limits** Negative infinity **Continuity** **Continuity** of essential functions Example Continuous on an ...

... The student will learn about: **continuity**, **continuity** properties, **and** solving inequalities using **continuity** properties. **Continuity** As we ... Objectives for Section 10.1 Introduction to **Limits** Functions **and** Graphs A Brief Review Analyzing a Limit **Limits** **One**-**Sided** **Limits** Example ...

Two-**sided** **and** **one**-**sided** **limits**: IV. Squeeze theorem: Let’s work through a few: Now you try this **one**... **Continuity** Intermediate value theorem Maple graph We had f(1.75) < 0 **and** f(2) > 0. So the root is between 1.75 **and** 2. Try the average, x = 1.875 Derivatives Derivatives (continued) ...

Chapter 3 **Limits** **and** **Continuity** 3.1 Derivative of a Function Quick Review Quick Review Quick Review Solutions Quick Review Solutions What you’ll learn about Definition of a Derivative Notation Relationship between the Graphs of f **and** f ' Graphing the Derivative from Data **One**-**sided** ...