Precalculus – MAT 129 Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF

Chapter Eleven Limits and an Introduction to Calculus

Ch. 11 Overview Introduction to Limits Techniques for Evaluating Limits The Tangent Line Problem Limits at Infinity and Limits of Sequences The Area Problem

11.1 – Introduction to Limits Definition of Limit Limits That Fail to Exist Properties of Limits and Direct Substitution

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:

11.1 – Limits That Fail to Exist There are three conditions under which limits do not exist: 1.The fxn approaches a different number coming from the right hand side as opposed to the left hand side. 2.The fxn heads off to pos./neg. infinity. 3.The fxn oscillates between two fixed values as x approaches c.

11.1 – Properties of Limits and Direct Substitution The basic limits and properties of limits are in the blue boxes on pg 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.

11.2 – Techniques for Evaluating Limits Dividing out Technique Rationalizing Technique One-sided Limits A Limit from Calculus

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.

Example 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

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.

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

11.2 – One-Sided Limits When the fxn approaches a different number on either side you need to do two different limits.

Example Pg. 795 Example 6 One-sided limits

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

11.3 – The Tangent Line Problem Slope and the Limit Process The Derivative of a Function

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. It’s really just a new slope formula.

Example Pg. 804 Example 4 Don’t be afraid of this new way to think of things.

10.3 – The Derivative of a Function See the blue box on pg 806. Derivative – using difference quotient

Example Pg. 807 Example 7

11.4 – Limits at Infinity and Limits of Sequences Limits at Infinity and Horizontal Asymptotes Limits of Sequences

11.4 – Limits at Infinity and Horizontal Asymptotes Follow the blue boxes and examples underneath.

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.

11.5 – The Area Problem Finding the limit of a summation. The Area Problem

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.