INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences  2011 Pearson Education, Inc. Chapter 10 Limits and Continuity

 2011 Pearson Education, Inc. 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

 2011 Pearson Education, Inc. Limits Limits (Continued) Continuity Continuity Applied to Inequalities 10.1) 10.2) 10.3) Chapter 10: Limits and Continuity Chapter Outline 10.4)

 2011 Pearson Education, Inc. 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:

 2011 Pearson Education, Inc. Chapter 10: Limits and Continuity 10.1 Limits Properties of Limits for any positive integer n

 2011 Pearson Education, Inc. Chapter 10: Limits and Continuity 10.1 Limits Example 3 – Applying Limit Properties 1 and 2 Properties of Limits

 2011 Pearson Education, Inc. Chapter 10: Limits and Continuity 10.1 Limits Example 5 – Limit of a Polynomial Function Find an expression for the polynomial function, Solution: where

 2011 Pearson Education, Inc. 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

 2011 Pearson Education, Inc. 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.

 2011 Pearson Education, Inc. 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

 2011 Pearson Education, Inc. Chapter 10: Limits and Continuity 10.2 Limits (Continued) Example 5 – Limits at Infinity for Polynomial Functions Find the limit (if it exists). Solution:

 2011 Pearson Education, Inc. 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:

 2011 Pearson Education, Inc. 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.

 2011 Pearson Education, Inc. 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.

 2011 Pearson Education, Inc. 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.

 2011 Pearson Education, Inc. 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.

 2011 Pearson Education, Inc. 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,∞).

 2011 Pearson Education, Inc. 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,∞).