Homework Homework Assignment #4 Read Section 2.5 Page 91, Exercises: 1 – 33 (EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page Find the points of discontinuity and state whether f (x) is left- or right-continuous, or neither at these points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page (a) For the function shown, determine the one- sided limits at the points of discontinuity. (b) Which of the discontinuities is removable and how should f be redefined to make it continuous at this point. The discontinuity at x = 2 is removable by defining f (2) = 6 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 91 Use the Laws of Continuity and Theorems 2–3 to show that the function is continuous. By Theorems 2 and 3, respectively, x and sin x are continuous. By Continuity Law ii, 3x and 4 sin x are continuous, and by Continuity Law i, 3x + 4 sin x is continuous Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 91 Use the Laws of Continuity and Theorems 2–3 to show that the function is continuous. By Theorem 3, 3 x and 4 x are continuous. By Theorem 2, x is continuous and by Continuity Law iv, f (x) is continuous Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 91 Determine the points at which the function is discontinuous and state the type of discontinuity. There is an infinite discontinuity at x = 0. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 91 Determine the points at which the function is discontinuous and state the type of discontinuity. There is a jump discontinuity at each integer value of x. The function is right-continuous at each jump discontinuity. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 91 Determine the points at which the function is discontinuous and state the type of discontinuity. The function is not defined for x < 0 and it is right- continuous at x = 0. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 91 Determine the points at which the function is discontinuous and state the type of discontinuity. The function has a jump discontinuity at x = 2, where it is neither right- nor left-continuous. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework, Page 91 Determine the points at which the function is discontinuous and state the type of discontinuity. The function is continuous for all values of x. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 2: Limits Section 2.5: Evaluating Limits Algebraically Jon Rogawski Calculus, ET First Edition

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Examining the graph in Figure 1, it is apparent that the value of f(x) approaches 8 as x approaches 4. In this section we will look at algebraic methods for evaluating such limits. We can not use substitution in this case as substitution yields:

Indeterminate Forms The function f (x) has an indeterminate form at x = c if, when f (x) is evaluated at x = c, we obtain an undefined expression of the type: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The function f (x) is also indeterminate at x = c. If possible, transform f (x) algebraically into a new expression that is defined and continuous at x = c.

Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 97 Evaluate the limit or state that it does not exist. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In some cases, such as that shown in Figure 2, the limit at a given point does not exist because the right– and left–hand limits are not equal.

Homework Homework Assignment #5 Read Section 2.6 Page 97, Exercises: 1 – 49 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company