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Let $y = f (x) = \frac{x^{4} + 4 x^{3} + 2 x^{2}}{x^{3} + x}$ . Find all the asymptotes of the function, it they exist.

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Answer

$The function has a vertical asymptote at x = 0, and there is no horizontal asymptote$

Steps

Step 1 ：Find the zeros of the denominator: $x^{3} + x = x (x^{2} + 1) = 0$

Step 2 ：The denominator has one real zero at $x = 0$

Step 3 ：Find the limit of the function as x approaches infinity: $lim_{x \to \infty} \frac{x^{4} + 4 x^{3} + 2 x^{2}}{x^{3} + x}$

Step 4 ：The limit as x approaches infinity is infinity, so there is no horizontal asymptote

Step 5 ：Find the limit of the function as x approaches 0: $lim_{x \to 0} \frac{x^{4} + 4 x^{3} + 2 x^{2}}{x^{3} + x}$

Step 6 ：The limit as x approaches 0 is 0

Step 7 ： $The function has a vertical asymptote at x = 0, and there is no horizontal asymptote$