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Find all horizontal and vertical asymptotes (if any). t(x)={x²}{x²}-x-6}

User Mbigun
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Final answer:

To find the horizontal and vertical asymptotes of the function t(x) = (x²)/(x² - x - 6), we analyze the behavior of the function as x approaches positive or negative infinity. The vertical asymptotes are x = 3 and x = -2, and the horizontal asymptote is y = 1.

Step-by-step explanation:

To find the horizontal and vertical asymptotes of the function t(x) = (x²)/(x² - x - 6), we need to analyze the behavior of the function as x approaches positive or negative infinity.

Vertical Asymptotes: The vertical asymptotes occur when the denominator of the function is equal to zero. So, we need to find the roots of the denominator equation x² - x - 6 = 0.

By factoring the quadratic equation, we get (x - 3)(x + 2) = 0. Therefore, x = 3 and x = -2. These are the vertical asymptotes of the function.

Horizontal Asymptotes: For rational functions, we compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0. If the degree of the numerator is equal to the degree of the denominator, there is a horizontal asymptote at the ratio of the leading coefficients. In this case, both the numerator and denominator have a degree of 2, so the horizontal asymptote is y = 1.

User Nickcen
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