Q. 61 61. Can the graph of a rational ... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 4: Q. 61 (page 226)

61. Can the graph of a rational function have both a horizontal and an oblique asymptote? Explain.

Short Answer

Expert verified

If $m > p$ or $m < p$ there will be no perfect division occurs, it may have or have not oblique asymptote and it will not have a horizontal asymptote.

Hence, the graph of a rational function can not have both a horizontal and an oblique asymptote.

Step by step solution

Step 1 Let us assume a function.

$f (x) = \frac{a x^{m} + b x^{m - 1} + . . . . . + k}{a_{1} x^{p} + b_{1} x^{p - 1} + . . . . + k_{1}}$ where $k, k_{1}$ are the constants.

Let $m$ be the degree of the numerator.

$p$ be the degree of the denominator.

If $m < p$ or $m > p$ , then there will be no perfect division occurs.

Step 2 If m<p then limx→∞fx becomes 0as a x is left undivided in the denominator.

If $m 鈮� p + 2$ then the quotient obtained is a polynomial of degree $2$ or higher then this may or may not have a oblique asymptote but it cannot have any horizontal asymptote as no perfect long division occurs.

Hence a single rational function cannot have horizontal as well as oblique asymptote at the same time.

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91影视

61. Can the graph of a rational function have both a horizontal and an oblique asymptote? Explain.

Short Answer

Step by step solution

Step 1 Let us assume a function.

Step 2 If m<p then limx→∞fx becomes 0as a x is left undivided in the denominator.

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