Explanation Formula used: Indeterminate form: The ratio of two functions in which both of these functions tend to zero in the limit. Calculation: Part a: Let the limit function is f ( x ) = x 2 − 1 x − 1 . The limit leads to an indeterminate form at x = 1 . Numerator of the function is x 2 − 1 . Substitute x = 1 in the numerator. x 2 − 1 = 1 2 − 1 = 0 Denominator of the function is x − 1 . Substitute x = 1 in the denominator. x − 1 = 1 − 1 = 0 Thus, the limit of the function is 0 0 . Now, check the right- and left-hand limits at x = 1 . lim x → 1 − x 2 − 1 x − 1 = lim x → 1 − ( x − 1 ) ( x + 1 ) x − 1 = lim x → 1 − ( x + 1 ) = 1 + 1 = 2 lim x → 1 + x 2 − 1 x − 1 = lim x → 1 + ( x − 1 ) ( x + 1 ) x − 1 = lim x → 1 + ( x + 1 ) = 1 + 1 = 2 Hence, the given statement is false. Part b: Let the limit function is f ( x ) = x 2 − 1 x − 1

4th Edition

ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 2.5, Problem 2PQ

To determine

Give the counterexample to shoe that these statements are false:

a. If f(c) is indeterminate, then the right- and left-hand limits as x→c are not equal.

b. If limx→cf(x) exists, then f(c) is not indeterminate.

c. If f(x) is undefined at x=c, then f(x) has an indeterminate form at x=c

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Chapter 2.5, Problem 1PQ

Chapter 2.5, Problem 3PQ

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i.) lim x→1 x² + x -2/x - 1; a. equals 3 b. equals 5 c. equals 4 d. none of the above/other answers are correct ii.) If f(x) = 1, the limit of f(x) as x approahes any real number: a. equals zero because f(x) is a constant b. equals 1 provided that the domain of f(x) is xER c. two of the other answers are correct d. does not exist

15. Vlad and Karl discuss about the continuity of the given function. Vlad asserts that the function is continuous at x=-1 while Karl claims it is not. Who among the two is correct? * 1 -, x -1 O Karl O Vlad O Both of them None of them

If f(x) = (3x – 2)10 , then f(1+ h) – f(1) lim h→0 h O 15 O 5 O 7.5 O 10 O 30 O 20

Chapter 2 Solutions

EBK CALCULUS

Ch. 2.1 - Prob. 1PQ Ch. 2.1 - Prob. 2PQ Ch. 2.1 - Prob. 3PQ Ch. 2.1 - Prob. 4PQ Ch. 2.1 - Prob. 1E Ch. 2.1 - Prob. 2E Ch. 2.1 - Prob. 3E Ch. 2.1 - Prob. 4E Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E

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Solution Summary: The author explains that if f(c) is indeterminate, then the right- and left-hand limits are not equal.

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Give the counterexample to shoe that these statements are false: a. If f(c) is indeterminate, then the right- and left-hand limits as x→c are not equal. b. If limx→cf(x) exists, then f(c) is not indeterminate. c. If f(x) is undefined at x=c, then f(x) has an indeterminate form at x=c

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Chapter 2 Solutions

Give the counterexample to shoe that these statements are false:

a. If f(c) is indeterminate, then the right- and left-hand limits as x→c are not equal.

b. If limx→cf(x) exists, then f(c) is not indeterminate.

c. If f(x) is undefined at x=c, then f(x) has an indeterminate form at x=c