(a) Explanation Let us define a function f : R → R f ( x ) = ( 1 + x ) n Now, let us define an interval [ 0 , x ] .Since, f ( x ) = ( 1 + x ) n is continuous on ( 0 , x ) and differentiable on [ 0 , x ] . It is evident from the function, f ( x ) = ( 1 + x ) n that f ( 0 ) = 1 From Mean Value Theorem (b) To determine lim x → 0 ( ( 1 + x ) n − 1 x )

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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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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Question

Chapter 4.2, Problem 44PS

(a)

To determine

The proof that there is a number ‘c’ between 0 and ‘x’ for which (1+x)n−1x=n(1+c)n−1

(b)

To determine

limx→0( (1+x)n−1x)

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Chapter 4.2, Problem 43PS

Chapter 4.2, Problem 45PS

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Chapter 4 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

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The proof that there is a number ‘c’ between 0 and ‘x’ for which ( 1 + x ) n − 1 x = n ( 1 + c ) n − 1

Solution Summary: The author explains the proof that there is a number ‘c’ between 0 and ‘x’ for which (1+x)n is continuous and differentiable.