Explanation Theorem used: Limits of the polynomial at infinity: “If p ( x ) = a n x n + a n − 1 x n − 1 + ... + a 0 , then lim x → ∞ p ( x ) = lim x → ∞ a n x n = ± ∞ and lim x → − ∞ p ( x ) = lim x → − ∞ a n x n = ± ∞ ”. Each limit will be either − ∞ or ∞ , depending on n and a n . (1). A polynomial of degree zero is a constant function p ( x ) = a 0 and its limit as x approaches to ∞ or − ∞ is the number a 0 . (2). For any polynomial of degree 1 or greater, the theorem states that the limit x as approaches ∞ or − ∞ cannot be equal to number. Calculation: Consider the limits at infinity in the polynomial function p ( x ) . From the above theorem, the limiting value of p ( x ) depends upon the value of a n and n . Check the condition by changing the values of n and a n as follows. Here, the degree of any polynomial should not negative, so that the value of n is always greater than or equal to 0. Take n = 0 and a n = 0 and apply the limit as x tends to infinity in p ( x ) . lim x → ∞ p ( x ) = lim x → ∞ ( a n x n + a n − 1 x n − 1 + ... + a 0 ) = lim x → ∞ a n x n ( By above theorem ) = 0 ( ∵ n = 0 and a n = 0 ) Thus, for n = 0 and any value of a n , the lim x → ∞ p ( x ) will be a n . Take n = 1 and a n = 1 and apply limit as x tends to infinity in p ( x ) . lim x → ∞ p ( x ) = lim x → ∞ ( a n x n + a n − 1 x n − 1 +

Additional Calculus Topics for Calculus for Business, Economics, Life Sciences and Social Sciences

13th Edition

ISBN: 9780321931696

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Publisher: PEARSON

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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 2.2, Problem 83E

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To discuss: The conditions under n and an which satisfy the limit of the polynomial function anxn+an−1xn−1+...+a0 to be ∞ and −∞.

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Chapter 2.2, Problem 82E

Chapter 2.2, Problem 84E

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Additional Calculus Topics for Calculus for Business, Economics, Life Sciences and Social Sciences

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The conditions under n and a n which satisfy the limit of the polynomial function a n x n + a n − 1 x n − 1 + ... + a 0 to be ∞ and − ∞ .

Solution Summary: The author explains that conditions under n and a_n satisfy the limit of the polynomial function.

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To discuss: The conditions under n and an which satisfy the limit of the polynomial function anxn+an−1xn−1+...+a0 to be ∞ and −∞.

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Chapter 2 Solutions