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Here's how to find slant asymptotes in rational functions:

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Conditions for a Slant Asymptote:

A rational function will have a slant asymptote only if the degree of the numerator (highest exponent in the numerator) is exactly one greater than the degree of the denominator (highest exponent in the denominator).

Steps to Find the Slant Asymptote:

  1. Check the Degrees: Make sure the condition above is met. If the numerator's degree is less than or equal to the denominator's degree, there won't be a slant asymptote (you might have a horizontal asymptote instead).

  2. Long Division: Perform long division with the numerator as the dividend and the denominator as the divisor.

  3. Slant Asymptote as the Quotient: The quotient you get from the long division (ignoring the remainder) represents the equation of the slant asymptote.


Let's find the slant asymptote of the function f(x) = (x^2 + 5x + 2) / (x + 3).

  • Degrees: The numerator (x^2) has a degree of 2, and the denominator (x) has a degree of 1. Since 2 is one greater than 1, we can find a slant asymptote.

  • Long Division: Perform long division. You'll end up with a quotient of x + 2 and a remainder of -4.

  • Slant Asymptote: The quotient (x + 2) represents the equation of the slant asymptote. Therefore, y = x + 2 is the slant asymptote of f(x).

Additional Resources:

You can find more explanations and visual examples of slant asymptotes online at [various resources on slant asymptotes].

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