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Question Write the equation of all horizontal asymptotes of the function f(x)=(6x-e^(x))/(3x-2x^(2)).

Write the equation of all horizontal asymptotes of the function f(x)=6x−ex3x−2x2f(x)=\frac{6x-e^{x}}{3x-2x^{2}}.

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Full solution

Q. Write the equation of all horizontal asymptotes of the function f(x)=6x−ex3x−2x2f(x)=\frac{6x-e^{x}}{3x-2x^{2}}.
  1. Question Prompt: Question Prompt: Determine the equation of all horizontal asymptotes for the function f(x)=6x−ex3x−2x2f(x) = \frac{6x - e^x}{3x - 2x^2}.
  2. Analyze Degrees: Analyze the degrees of the polynomials in the numerator and the denominator. The numerator has terms 6x6x and exe^x, where exe^x grows faster than any polynomial. The denominator has terms 3x3x and −2x2-2x^2, with the highest degree being 22.
  3. Determine Asymptote: Since the highest degree in the denominator (22) is greater than the highest degree in the numerator (11, considering exe^x as exponential growth), the horizontal asymptote is y=0y = 0. This is because as xx approaches infinity, the polynomial term of higher degree in the denominator dominates, making the fraction approach zero.

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