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Chapter 2: Problem 10

Find all vertical and horizontal asymptotes of the graph of the function. $$f(x)=\frac{1}{(x-2)^{3}}$$

Short Answer

Expert verified
The vertical asymptote is at x=2 and the horizontal asymptote is at y=0.

Step by step solution

01

Finding Vertical Asymptotes

A vertical asymptote of a function occurs where the function is undefined. For the given function \( f(x)=\frac{1}{(x-2)^3} \), this happens when the denominator is zero. Set the denominator equal to zero and solve for x: \((x-2)^3 = 0\), \( x=2 \). This means we have a vertical asymptote at \(x=2\).
02

Finding Horizontal Asymptotes

To find horizontal asymptotes, we examine the limit of the function as x approaches positive and negative infinity. For the given function, as x approaches positive infinity, the value of the function approaches 0, and as x approaches negative infinity, the value of the function also approaches 0. Therefore the horizontal asymptote is at \(y=0\).

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Most popular questions from this chapter

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-3 x^{3}-x^{2}-12 x-20\) (Hint: One factor is \(\left.x^{2}+4 .\right)\)

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