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Chapter 2: Problem 14

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

Short Answer

Expert verified
The vertical asymptote is x = -2 and there are no horizontal asymptotes.

Step by step solution

01

Finding Vertical Asymptotes

Set the denominator equal to zero and solve for x. So solve the equation, \(x+2 = 0\). This will give x = -2. So, x = -2 is a vertical asymptote.
02

Confirming no Undefined Regions

An additional step is to make sure that no other x-values make the function undefined. In this case, x = -2 is the only value that causes the denominator to be zero, so there are no other undefined regions.
03

Finding Horizontal Asymptotes

Investigate the limit of the function as x approaches infinity and negative infinity. For this function, both \(\lim_{x\to \infty} f(x)\) and \(\lim_{x\to -\infty} f(x)\) result in 4x. Since the coefficient of x in the numerator is larger than the coefficient of x in the denominator, the rate at which the function increases outpaces the rate at which x approaches infinity. Hence, there are no horizontal asymptotes.

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