homework homework assignment #4 read section 2.5 page 91, exercises: 1 – 33 (eoo) quiz next time...
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Homework
Homework Assignment #4 Read Section 2.5 Page 91, Exercises: 1 – 33 (EOO) Quiz next time
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 912. Find the points of discontinuity and state whether f (x) is left- or right-continuous, or neither at these points.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
x
y
Homework, Page 915. (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 CalculusCopyright © 2008 W. H. Freeman and Company
x
y
6
2 4-2
0 0
2
lim , lim 2
lim 6x x
x
f x f x
f x
Homework, Page 91Use 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 CalculusCopyright © 2008 W. H. Freeman and Company
9. 3 4sinf x x x
Homework, Page 91Use the Laws of Continuity and Theorems 2–3 to show that the function is continuous.
By Theorem 3, 3x and 4x are continuous. By Theorem 2, 1 + 4x is continuous and by Continuity Law iv, f (x) is continuous
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
313.
1 4
x
xf x
Homework, Page 91Determine the points at which the function is discontinuous and state the type of discontinuity.
There is an infinite discontinuity at x = 0.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
117. f x
x
Homework, Page 91Determine 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 CalculusCopyright © 2008 W. H. Freeman and Company
21. f x x
Homework, Page 91Determine 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 CalculusCopyright © 2008 W. H. Freeman and Company
3 3225. 3 9f x x x
Homework, Page 91Determine 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 CalculusCopyright © 2008 W. H. Freeman and Company
2 3 2
29. x x
f xx
Homework, Page 91Determine the points at which the function is discontinuous and state the type of discontinuity.
The function is continuous for all values of x.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
33. tan sinf x x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Chapter 2: LimitsSection 2.5: Evaluating Limits Algebraically
Jon Rogawski
Calculus, ET First Edition
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Examining the graph in Figure 1, it is apparent thatthe value of f(x) approaches8 as x approaches 4. In this section we will look at algebraic methods forevaluating such limits.
2
4
16 16 16 0lim
4 4 4 0x
x
x
0and is undefined as division by zero is undefined.
0
We can not use substitution in this case as substitutionyields:
Indeterminate FormsThe 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 CalculusCopyright © 2008 W. H. Freeman and Company
0 00, , 0, , 1 , , 0
0
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 CalculusCopyright © 2008 W. H. Freeman and Company
2
2
646. lim
9x
x
x
Example, Page 97 Evaluate the limit or state that it does not exist.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3
2
48. lim
2x
x x
x
Example, Page 97 Evaluate the limit or state that it does not exist.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3
2
210. lim
2x
x x
x
Example, Page 97 Evaluate the limit or state that it does not exist.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
16
416. lim
16x
x
x
Example, Page 97 Evaluate the limit or state that it does not exist.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
20
1 126. lim
x x x x
Example, Page 97 Evaluate the limit or state that it does not exist.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2
30. lim sec tan
Rogawski CalculusCopyright © 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.
2 2
22 2
lim , lim
lim lim lim D.N. E.x x
xx x
f x f x
f x f x f x
Homework
Homework Assignment #5 Read Section 2.6 Page 97, Exercises: 1 – 49 (EOO)
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company