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Chapter 2: Problem 13

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

Short Answer

Expert verified
The vertical asymptotes are at \(x = -1\) and \(x = 1\). There is no horizontal asymptote.

Step by step solution

01

Finding the Vertical Asymptotes

To find the vertical asymptotes, set the denominator of the function equal to zero and solve for x. So, solve \(x^{2}-1=0\). From this, we can find the following two solutions: \(x = -1, 1\). So, the vertical asymptotes are the lines \(x = -1\) and \(x = 1\).
02

Finding the Horizontal Asymptotes

For the horizontal asymptotes, we need to consider the limit of the function as x goes to negative and positive infinity. Consider \(lim_{x\rightarrow\infty} f(x)\) and \(lim_{x\rightarrow -\infty} f(x)\). The function \(f(x)=\frac{x^{3}}{x^{2}-1}\) can be rewritten as \(f(x)=\frac{x}{1-\frac{1}{x^{2}}}\). With this simplification, it is clear that as x goes to positive or negative infinity, the fraction \(\frac{1}{x^{2}}\) goes to zero. Therefore, the function approaches \(x\) as it increases to positive or negative infinity. Thus, there is no horizontal asymptote.

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