INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences  2007 Pearson Education Asia Chapter 10 Limits and Continuity

 2007 Pearson Education Asia INTRODUCTORY MATHEMATICAL ANALYSIS 0.Review of Algebra 1.Applications and More Algebra 2.Functions and Graphs 3.Lines, Parabolas, and Systems 4.Exponential and Logarithmic Functions 5.Mathematics of Finance 6.Matrix Algebra 7.Linear Programming 8.Introduction to Probability and Statistics

 2007 Pearson Education Asia 9.Additional Topics in Probability 10.Limits and Continuity 11.Differentiation 12.Additional Differentiation Topics 13.Curve Sketching 14.Integration 15.Methods and Applications of Integration 16.Continuous Random Variables 17.Multivariable Calculus INTRODUCTORY MATHEMATICAL ANALYSIS

 2007 Pearson Education Asia To study limits and their basic properties. To study one-sided limits, infinite limits, and limits at infinity. To study continuity and to find points of discontinuity for a function. To develop techniques for solving nonlinear inequalities. Chapter 10: Limits and Continuity Chapter Objectives

 2007 Pearson Education Asia Limits Limits (Continued) Continuity Continuity Applied to Inequalities 10.1) 10.2) 10.3) Chapter 10: Limits and Continuity Chapter Outline 10.4)

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.1 Limits Example 1 – Estimating a Limit from a Graph The limit of f(x) as x approaches a is the number L, written as a. Estimate lim x→1 f (x) from the graph. Solution: b. Estimate lim x→1 f (x) from the graph. Solution:

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.1 Limits Properties of Limits for any positive integer n

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.1 Limits Example 3 – Applying Limit Properties 1 and 2 Properties of Limits

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.1 Limits Example 5 – Limit of a Polynomial Function Find an expression for the polynomial function, Solution: where

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.1 Limits Example 7 – Finding a Limit Example 9 – Finding a Limit Find. Solution: If,find. Solution: Limits and Algebraic Manipulation If f (x) = g(x) for all x  a, then

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.2 Limits (Continued) Example 1 – Infinite Limits Infinite Limits Infinite limits are written as and. Find the limit (if it exists). Solution: a. The results are becoming arbitrarily large. The limit does not exist. b. The results are becoming arbitrarily large. The limit does not exist.

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.2 Limits (Continued) Example 3 – Limits at Infinity Find the limit (if it exists). Solution: a.b. Limits at Infinity for Rational Functions If f (x) is a rational function, and

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.2 Limits (Continued) Example 5 – Limits at Infinity for Polynomial Functions Find the limit (if it exists). Solution:

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.3 Continuity Example 1 – Applying the Definition of Continuity Definition f(x) is continuous if three conditions are met: a. Show that f(x) = 5 is continuous at 7. Solution: Since,. b. Show that g(x) = x 2 − 3 is continuous at −4. Solution:

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.3 Continuity Example 3 – Discontinuities a. When does a function have infinite discontinuity? Solution: A function has infinite discontinuity at a when at least one of the one-sided limits is either ∞ or −∞ as x →a. b. Find discontinuity for Solution: f is defined at x = 0 but lim x→0 f (x) does not exist. f is discontinuous at 0.

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.3 Continuity Example 5 – Locating Discontinuities in Case-Defined Functions For each of the following functions, find all points of discontinuity.

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.3 Continuity Example 5 – Locating Discontinuities in Case-Defined Functions Solution: a. We know that f(3) = = 9. Because and, the function has no points of discontinuity.

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.3 Continuity Example 5 – Locating Discontinuities in Case-Defined Functions Solution: b. It is discontinuous at 2, lim x→2 f (x) exists.

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.4 Continuity Applied to Inequalities Example 1 – Solving a Quadratic Inequality Solve. Solution: Let. To find the real zeros of f, Therefore, x 2 − 3x − 10 > 0 on (−∞,−2)  (5,∞).

 2007 Pearson Education Asia Chapter 10: Limits and Continuity 10.4 Continuity Applied to Inequalities Example 3 – Solving a Rational Function Inequality Solve. Solution: Let. The zeros are 1 and 5. Consider the intervals: (−∞, 0) (0, 1) (1, 5) (5,∞) Thus, f(x) ≥ 0 on (0, 1] and [5,∞).