Answered: 2x - 8(x)^1/3

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter4: Polynomial And Rational Functions

Section4.6: Rational Functions

Problem 4SC

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Question

How can I find the oblique asymptote of the equation 2x - 8(x)^1/3

Expert Solution

Step 1

Given:

$2 x - 8 \sqrt[3]{x}$

To find:

The oblique or slant asymptote.

Step 2

The Line $y = m x + b$ is slant asymptote of the function $y = f (x)$ ,

If either $m = \lim_{x \to \infty} (\frac{f (x)}{x}) = L$ or $m = \lim_{x \to - \infty} (\frac{f (x)}{x}) = L$ , and L is finite and nonzero.

calculate the first limit,

$\lim_{x \to \infty} (\frac{- 8 \sqrt[3]{x} + 2 x}{x})$ ,

$\lim_{x \to \infty} (\frac{- 8 \sqrt[3]{x} + 2 x}{x}) = (2 \lim_{x \to \infty} \frac{- 4 \sqrt[3]{x} + x}{x})$

Multiply and divide by x.

$\lim_{x \to \infty} (\frac{- 8 \sqrt[3]{x} + 2 x}{x}) = (2 \lim_{x \to \infty} \frac{x \frac{- 4 \sqrt[3]{x} + x}{x}}{x})$

$\lim_{x \to \infty} (\frac{- 8 \sqrt[3]{x} + 2 x}{x}) = (2 \lim_{x \to \infty} 1 - \frac{4}{x^{\frac{2}{3}}})$

$\lim_{x \to \infty} (\frac{- 8 \sqrt[3]{x} + 2 x}{x}) = 2 - 2 (4 \lim_{x \to \infty} \frac{1}{x^{\frac{2}{3}}})$

$\lim_{x \to \infty} (\frac{- 8 \sqrt[3]{x} + 2 x}{x}) = 2 - 8 (\lim_{x \to \infty} \frac{1}{x^{\frac{2}{3}}})$

$\lim_{x \to \infty} (\frac{- 8 \sqrt[3]{x} + 2 x}{x}) = 2 - 8 (0)$

$\lim_{x \to \infty} (\frac{- 8 \sqrt[3]{x} + 2 x}{x}) = 2$

Since the value of the limit is finite and non zero,

Then there is slant asymptote with m=2.

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