Precalculus

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Precalculus MAT 129

• Instructor Rachel Graham
• Location BETTS Rm. 107
• Time 8 1120 a.m. MWF

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Chapter Eleven

• Limits and an Introduction to Calculus

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Ch. 11 Overview

• Introduction to Limits
• Techniques for Evaluating Limits
• The Tangent Line Problem
• Limits at Infinity and Limits of Sequences
• The Area Problem

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11.1 Introduction to Limits

• Definition of Limit
• Limits That Fail to Exist
• Properties of Limits and Direct Substitution

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11.1 Definition of Limit

• If f(x) becomes arbitrarily close to a unique
  number L as x approaches c from either side, the
  limit of f(x) as x approaches c is L.
• This is written as

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11.1 Limits That Fail to Exist

• There are three conditions under which limits do
  not exist
• The fxn approaches a different number coming from
  the right hand side as opposed to the left hand
  side.
• The fxn heads off to pos./neg. infinity.
• The fxn oscillates between two fixed values as x
  approaches c.

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11.1 Properties of Limits and Direct
Substitution

• The basic limits and properties of limits are in
  the blue boxes on pg. 785.
• For direct substitution you take the c in the
  limit and substitute it into the function. If you
  get a number that is the limit. If you get 0 or
  0/0 or /0 you have to use some other method to
  find the limit.

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11.2 Techniques for Evaluating Limits

• Dividing out Technique
• Rationalizing Technique
• One-sided Limits
• A Limit from Calculus

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11.2 Dividing out Technique

• If you factor either the top or the bottom or
  both of the rational polynomial and then cancel.
  You can then use direct substitution to solve the
  limit.

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Example 1.11.2

• Pg. 791 Example 1
• This is the Dividing Out Technique
• Begin by factoring any polynomials that can
• Divide out common factors
• Use direct substitution to find the limit

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11.2 Rationalizing Technique

• If you rationalize (take top and bottom times the
  conjugate) the rational polynomial and then
  cancel. You can then use direct substitution to
  solve the limit.

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Example 2.11.2

• Pg. 793 Example 3
• This is the rationalizing technique
• Rationalize either the numerator or the
  denominator
• Divide out the common factor
• Use direct substitution

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11.2 One-Sided Limits

• When the fxn approaches a different number on
  either side you need to do two different limits.

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Example 3.11.2

• Pg. 795 Example 6
• One-sided limits

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Example 4.11.2

• Pg. 797 Example 9
• Here we see the difference quotient again.

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11.3 The Tangent Line Problem

• Slope and the Limit Process
• The Derivative of a Function

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11.3 Slope and the Limit Process

• Page 801 803 do a really good job of bridging
  the gap between what we have learned so far and
  where we are going. Its really just a new slope
  formula.

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Example 1.11.3

• Pg. 804 Example 4
• Dont be afraid of this new way to think of
  things.

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10.3 The Derivative of a Function

• See the blue box on pg 806.
• Derivative using difference quotient

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Example 2.11.3

• Pg. 807 Example 7

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11.4 Limits at Infinity and Limits of Sequences

• Limits at Infinity and Horizontal Asymptotes
• Limits of Sequences

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11.4 Limits at Infinity and Horizontal
Asymptotes

• Follow the blue boxes and examples underneath.

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11.4 Limits of Sequences

• This topic leads us right into the heart of the
  area issue that is the heart of calculus.
• All that you do it to look at the behavior of a
  sequence as it goes on.
• If the sequence does not converge on a certain
  number we say that it diverges.

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11.5 The Area Problem

• Finding the limit of a summation.
• The Area Problem

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11.5 The Area Problem

• Again for this whole section we should go over as
  many examples in the section as possible.
• REMEMBER!!! All we are doing is coloring in under
  the function and finding the area.