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Chapter 2: Problem 9

Find all vertical and horizontal asymptotes of the graph of the function. $$f(x)=\frac{4}{x^{2}}$$

Short Answer

Expert verified
The function \(f(x)=\frac{4}{x^{2}}\) has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\).

Step by step solution

01

Find the vertical asymptotes

Set the denominator of the function equal to zero and solve for \(x\). So we get \(x^{2} = 0\), which gives \(x = 0\). So, \(x = 0\) is a vertical asymptote.
02

Find the horizontal asymptotes

The horizontal asymptotes depend on the limit of the function as \(x\) approaches positive and negative infinity. Let's check the limits:\n If \(x\) approaches positive infinity, \(\displaystyle \lim_{x \to \infty} \frac{4}{x^{2}} = 0\), and if \(x\) approaches negative infinity, \(\displaystyle \lim_{x \to -\infty} \frac{4}{x^{2}} = 0\). Therefore, \(y = 0\) is a horizontal asymptote.

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Most popular questions from this chapter

Complete the following. \begin{aligned} &i^{1}=i \quad i^{2}=-1 \quad i^{3}=-i \quad i^{4}=1\\\ &i^{5}=\quad i^{6}=\quad i^{7}=\quad i^{8}=\\\ &i^{9}=\\\ &i^{10}=\quad i^{11}=\quad i^{12}= \end{aligned} What pattern do you see? Write a brief description of how you would find \(i\) raised to any positive integer power.

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$h(x)=4 x^{2}-8 x+3$$

Find all real zeros of the function. $$f(x)=4 x^{3}-3 x-1$$

Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=3 f(x)$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{3}-3 x^{2}+x+5$$
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