How to find horizontal asymptotes

General
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how to find horizontal asymptotes

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how to find horizontal asymptotes

Answer:

To find the horizontal asymptotes of a function, you need to determine what happens to the function as x approaches positive or negative infinity. Here are the steps to follow:

  1. Take the limit of the function as x approaches positive or negative infinity.

  2. If the limit is a finite number (i.e., it approaches a specific value), then this is the horizontal asymptote.

  3. If the limit does not exist (i.e., it approaches infinity or negative infinity), then there is no horizontal asymptote.

  4. If the limit is zero, you will need to perform further analysis to determine if there is a horizontal asymptote.

Here’s an example: Let’s say we want to find the horizontal asymptote of the function f(x) = (2x^3 + 5x^2 - 8x) / (3x^3 + 9x^2 - 2x - 6).

  1. Take the limit of the function as x approaches positive infinity: lim(x->∞) (2x^3 + 5x^2 - 8x) / (3x^3 + 9x^2 - 2x - 6) = lim(x->∞) (2 + 5/x - 8/x^2) / (3 + 9/x - 2/x^2 - 6/x^3) = 2/3
  2. Take the limit of the function as x approaches negative infinity: lim(x->-∞) (2x^3 + 5x^2 - 8x) / (3x^3 + 9x^2 - 2x - 6) = lim(x->-∞) (2 - 5/x - 8/x^2) / (3 - 9/x - 2/x^2 - 6/x^3) = 2/3

Since both limits are equal to 2/3, the horizontal asymptote of the function is y = 2/3