Techniques for Computing Limits: The Limit Laws

Slides:

Advertisements

Similar presentations

EVALUATING LIMITS ANALYTICALLY

Advertisements

Composition of functions constructing a function from 2 functions (g o f) = g[f(x)] –for all x in domain of f such that f(x) is in domain of g –f is applied.

1.3 Evaluating Limits Analytically

1.5 Continuity. Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without.

1.4 Calculating limits. Limit Laws Suppose that c is a constant and the limits and exist. Then.

LIMITS Calculating Limits Using the Limit Laws LIMITS In this section, we will: Use the Limit Laws to calculate limits.

1.3 EVALUATING LIMITS ANALYTICALLY. Direct Substitution If the the value of c is contained in the domain (the function exists at c) then Direct Substitution.

Derivatives of polynomials Derivative of a constant function We have proved the power rule We can prove.

Techniques for Computing Limits 2.3 Calculus 1. The limit of a constant IS the constant. No matter what “x” approaches Limit Laws.

Limits and Their Properties

1.3 Evaluating Limits Analytically Objectives: -Students will evaluate a limit using properties of limits -Students will develop and use a strategy for.

Finding Limits Analytically 1.3. Concepts Covered: Properties of Limits Strategies for finding limits The Squeeze Theorem.

EVALUATING LIMITS ANALYTICALLY (1.3) September 20th, 2012.

Rates of Change and Limits

Section 1.6 Calculating Limits Using the Limit Laws.

Warm up Warm up 5/16 1. Do in notebook Evaluate the limit numerically(table) and graphically.

Miss Battaglia AB/BC Calculus

DO NOW: Find. HW: Finish WKSH 2.1 and a half – Calculating Limits Using the Limit Laws.

In previous sections we have been using calculators and graphs to guess the values of limits. Sometimes, these methods do not work! In this section we.

AP Calculus Chapter 1, Section 3

Limit Laws Suppose that c is a constant and the limits lim f(x) and lim g(x) exist. Then x -> a Calculating Limits Using the Limit Laws.

Section 11-1: Properties of Exponents Property of Negatives:

1.6 Continuity CALCULUS 9/17/14. Warm-up Warm-up (1.6 Continuity-day 2)

Aim: Evaluating Limits Course: Calculus Do Now: Aim: What are some techniques for evaluating limits? Sketch.

1.3 Evaluating Limits Analytically. Warm-up Find the roots of each of the following polynomials.

Section 1.3 – Evaluating Limits Analytically. Direct Substitution One of the easiest and most useful ways to evaluate a limit analytically is direct substitution.

SECTION 2.2 Finding Limits Graphically & Numerically.

In your own words: What is a limit?.

Calculating Limits Using The Limit Laws. 2 Basic Limit Laws where n is a positive integer. y = c |a|a   (a, c) y = x |a|a   (a, a) where n.

We have used calculators and graphs to guess the values of limits.  However, we have learned that such methods do not always lead to the correct answer.

TODAY IN CALCULUS…  Warm Up: Review simplifying radicals  Learning Targets :  You will use special products and factorization techniques to factor polynomials.

2.1 Rates of Change and Limits. What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided.

Limits and Their Properties 1 Copyright © Cengage Learning. All rights reserved.

Chapter 1 Limits and Their Properties. Copyright © Houghton Mifflin Company. All rights reserved.21-2 Figure 1.1.

Limits and Their Properties 1 Copyright © Cengage Learning. All rights reserved.

2.3 Calculating Limits Using the Limit Laws. Properties of Limits where n is a positive integer. y = c |a|a   (a, c) y = x |a|a   (a, a)

LIMITS Calculating Limits Using the Limit Laws LIMITS In this section, we will: Use the Limit Laws to calculate limits.

2.3 Calculating Limits Using the Limit Laws LIMITS AND DERIVATIVES In this section, we will: Use the Limit Laws to calculate limits.

1.4 Continuity and One-Sided Limits Main Ideas Determine continuity at a point and continuity on an open interval. Determine one-sided limits and continuity.

Warm Up Compute the following by using long division.

2.3 - Calculating Limits Using The Limit Laws

EVALUATING LIMITS ANALYTICALLY (1.3)

1.3 Evaluating Limits Analytically

12.2 Finding Limits Algebraically

Sec. 1.3: Evaluating Limits Analytically

Rates of Change and Limits

Rational Zero Theorem Rational Zero Th’m: If the polynomial

Finding Limits Analytically

IF c is constant and lim f(x) and lim g(x) exist then…

Techniques for Computing Limits: The Limit Laws

1.3 Evaluating Limits Analytically

Evaluating Limits Analytically with Trig

Section 1.2 Exponents & Radicals

OUR GOAL Find: Sec 2.2: Limit of Function and Limit Laws

AP Calculus September 6, 2016 Mrs. Agnew

Limits and Their Properties

1.6 Continuity Objectives:

EVALUATING LIMITS ANALYTICALLY

1.3 Evaluating Limits Analytically

§2.5. Continuity In this section, we use limits to

CONTINUITY AND ONE-SIDED LIMITS

Finding Limits Graphically and Numerically

§1.3: Properties of Limits with Trigonometry

Evaluating Limits Analytically

Notes Over 6.6 Possible Zeros Factors of the constant

Zeros of polynomial functions

CONTINUITY AND ONE-SIDED LIMITS

2.3 Continuity.

Presentation transcript:

Techniques for Computing Limits: The Limit Laws 1.6 Techniques for Computing Limits: The Limit Laws

Computing Limits Basic Limits: For real numbers b and c, and positive integers n:

Properties of Limits

Properties of Limits (continued)

Methods for Computing Limits A. “Plug-Ins” (Direct substitution) Using these basic limits and properties of limits, we can prove that the limit at c of the following kinds of functions can be evaluated by direct substitution of c for x. Direct substitution will work for: Polynomial Functions Rational Functions with c in domain Radical Functions with c in domain Trigonometric Functions with c in its domain

Examples

Methods for Computing Limits B. Rational Functions with c not in domain (“Single Holes”)

-

Example 4

C. Functions with radicals, c not in domain:

D. Special Trigonometric Limits Theorems:

Examples:

The Squeeze Theorem

Example (using the Squeeze Theorem):