Explanation Given information: f ( x ) = − 5 x 3 − 4 x 2 + 8 6 x 2 + 3 x + 2 Calculation: To find the limit at infinity of rational function use the following steps: Divide the numerator and denominator of function f ( x ) by highest power of variable x that appears in the denominator. Apply the limits rule and rules for limits at infinity, for any positive n , lim x → ± ∞ 1 x n = 0 . In the function, f ( x ) = − 5 x 3 − 4 x 2 + 8 6 x 2 + 3 x + 2 . The highest power of variable x appears in the denominator is x 2 . Divide the numerator and denominator of f ( x ) by x 2 , − 5 x 3 − 4 x 2 + 8 6 x 2 + 3 x + 2 = − 5 x − 4 + 8 x 2 6 + 3 x + 2 x 2 . Therefore, lim x → ∞ − 5 x 3 − 4 x 2 + 8 6 x 2 + 3 x + 2 = lim x → ∞ − 5 x − 4 + 8 x 2 6 + 3 x + 2 x 2 . The rule for the limit of quotient lim x → ∞ [ f ( x ) g ( x ) ] = lim x → ∞ f ( x ) lim x → ∞ g ( x ) = A B , if B ≠ 0 . By using the quotient rule, lim x → ∞ − 5 x 3 − 4 x 2 + 8 6 x 2 + 3 x + 2 = lim x → ∞ ( − 5 x − 4 + 8 x 2 ) lim x → ∞ ( 6 + 3 x + 2 x 2 ) . Then by using sum or difference rule, lim x → ∞ [ f ( x ) ± g ( x ) ] = lim x → ∞ f ( x ) ± lim x → ∞ g ( x ) = A ± B Therefore, the lim x → ∞ − 5 x 3 − 4 x 2 + 8 6 x 2 + 3 x + 2 = lim x → ∞ ( − 5 x ) − lim x → ∞ ( 4 ) + lim x → ∞ ( 8 x 2 ) lim x → ∞ ( 6 ) + lim x → ∞ ( 3 x ) + lim x → ∞ ( 2 x 2 )

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ISBN: 9780321964038

Author: GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher: Pearson Addison Wesley,

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Chapter 3.1, Problem 58E

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The value of limx→ ∞−5x3−4x2+86x2+3x+2 if it exists use the properties to decide whether limit exists.

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Chapter 3.1, Problem 57E

Chapter 3.1, Problem 59E

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The value of lim x → ∞ − 5 x 3 − 4 x 2 + 8 6 x 2 + 3 x + 2 if it exists use the properties to decide whether limit exists.

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The value of limx→ ∞−5x3−4x2+86x2+3x+2 if it exists use the properties to decide whether limit exists.

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