Evaluating Limits Numerically & Intro into Algebraic

Slides:



Advertisements
Similar presentations
Sec. 1.2: Finding Limits Graphically and Numerically.
Advertisements

A Dash of Limits. Objectives Students will be able to Calculate a limit using a table and a calculator Calculate a limit requiring algebraic manipulation.
Section 1.2 – Finding Limits Graphically and Numerically
Section 1.2 – Finding Limits Graphically and Numerically
Thinking Mathematically Algebra: Graphs, Functions and Linear Systems 7.3 Systems of Linear Equations In Two Variables.
Section Finding Limits Graphically and Numerically.
Evaluating Limits Analytically
Finding Limits Graphically and Numerically An Introduction to Limits Limits that Fail to Exist A Formal Definition of a Limit.
1.3 Evaluating Limits Analytically Objectives: -Students will evaluate a limit using properties of limits -Students will develop and use a strategy for.
Calculus Section 1.1 A Preview of Calculus What is Calculus? Calculus is the mathematics of change Two classic types of problems: The Tangent Line Problem.
1.2 Finding Limits Graphically and Numerically
Lesson 15-1 Limits Objective: To calculate limits of polynomials and rational functions algebraically To evaluate limits of functions using a calculator.
Limits Numerically Warm-Up: What do you think the following limit equals? If you are unsure at least recall what a limit is and see if that helps direct.
Warm Up. Getting Started Use your calculator to graph the following equation: What do you notice about the graph? Look closely!!!! Change your window.
Objectives: To evaluate limits numerically, graphically, and analytically. To evaluate infinite limits.
1 § 1-4 Limits and Continuity The student will learn about: limits, infinite limits, and continuity. limits, finding limits, one-sided limits,
11.1 Finding Limits Graphically and Numerically
Ch 11.2: Techniques for Evaluating Limits. Dividing Out Technique Used when direct substitution gives you a zero in the numerator and denominator Steps:
Finding Limits Graphically and Numerically 2015 Limits Introduction Copyright © Cengage Learning. All rights reserved. 1.2.
1. What is Calculus? The mathematics of tangent lines, slopes, areas, volumes, arc lengths, and curvatures 2. Pre-Calculus vs Calculus The mathematics.
MCV4U The Limit of a function The limit of a function is one of the basic concepts in all of calculus. They arise when trying to find the tangent.
AIM : How do we find limits of a function graphically & numerically? Do Now: Graph the piecewise function. HW2.2b – p76-7 # 3, 9, 22, 39, 40, 48 HW2.2b.
What is a limit ? When does a limit exist? Continuity Continuity of basic algebraic functions Algebra of limits The case 0/0.
1.1 Preview of Calculus Objectives: -Students will understand what calculus is and how it compares with precalculus -Students will understand that the.
Evaluate limits analytically Special limits you need to know.
The foundation of calculus
What is “calculus”? What do you learn in a calculus class?
Sect.1.5 continued Infinite Limits
1.5 Infinite Limits Main Ideas
Evaluating Limits Analytically
2.1 Rates of Change and Limits
Chapter 10 Limits and the Derivative
U3 L2 Limits of Primary Trig Functions
Section Finding Limits Graphically and Numerically
1.2 Finding Limits Numerically and Graphically.
Evaluating Limits Analytically
Limits and Continuity The student will learn about: limits,
Warm up Warm up 1. Do in notebook
What is “calculus”? What do you learn in a calculus class?
1.2 Finding Limits Graphically and Numerically, part 1
What Do Limits Have To Do With Calculus?
Evaluating Limits Analytically
Ways to Evaluate Limits:
Finding Limits: An Algebraic Approach
Prep Book Chapter 3 - Limits of Functions
Graphs of Rational Functions
10.3 – More on Limits Take a few minutes to review your last test!!
Warmup: Let’s Practice Graphing Piecewise Functions Ourselves
2. Properties of Limits.
Parabolas Objective: Be able to identify the vertex, focus and directrix of a parabola and create an equation for a parabola. Thinking Skill: Explicitly.
What LIMIT Means Given a function: f(x) = 3x – 5 Describe its parts.
Warmup: Let’s Practice Graphing Piecewise Functions Ourselves
The Limit of a Function.
Composite functions.
What is “calculus”? What do you learn in a calculus class?
11.1 Intro to Limits.
Finding Limits A Graphical & Numerical Approach
2.3 Calculating Limits Using the Limit Laws
Limits Graphically & Limit Properties
Today in Precalculus Notes: Limits (no calculators) Go over quiz
2.2 The Limit of a Function.
3.5 Polynomial and Rational Inequalities
1.5: Limits We will learn about: limits, finding limits,
2.1 Introduction to Limits
4. Algebraic Limits.
Warm-up Enter the two functions into the y = in your
Example 1: Solving Rational Equations
More Definite Integrals
Presentation transcript:

Evaluating Limits Numerically & Intro into Algebraic TS: Explicitly assessing information and drawing conclusions Objective: To be able to evaluate a limit graphically and analytically.

General Limit A general limit exists on f (x) when x = c, if the left- and right-hand limits are both equal there. Mathematic Notation: In other words: f (x)  L as x  c

Limits Graphically 6 6 6 6

Limits Graphically Undefined 2 2 2

Limits Graphically 5 8 8 8

Limits Graphically – 3 3 7 DNE

Possible ways to evaluate a limit without a graph. Substitution Factoring Conjugate Table

value is the limit of the Finding Limits = 7 = 7 = 7 If a function approaches the same value from both directions, then that value is the limit of the function at that point. x y x y .9 6.71 1.1 7.31 .99 6.9701 1.01 7.0301 .999 6.997 1.001 7.003

Finding Limits = DNE or NL = 3 = –3 If the Left-hand limit and the Right-hand limit are not equal, the general limit does not exist. x y x y –1.1 3.1 –.9 –2.9 –1.01 3.01 –.99 –2.99 –1.001 3.001 –.999 –2.999

Finding Limits = DNE or NL = -∞ = ∞ x y x y If either the Left-hand limit, Right-hand limit, or both do not exist, the general limit will not exist. 2.9 –44.1 3.1 56.1 2.99 –494 3.01 506.01 2.999 –4994 3.001 5006

How should we best use our calculator to help us if we need to make a table?