What is how to find horizontal asymptotes?

Here's information on how to find horizontal asymptotes:

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To find <a href="https://www.wikiwhat.page/kavramlar/horizontal%20asymptotes">horizontal asymptotes</a> of a function, you primarily examine the function's behavior as x approaches positive and negative infinity. There are three general scenarios to consider:

1. Degree of Numerator < Degree of Denominator: If the degree (highest power of x) in the numerator is less than the degree in the denominator, the horizontal asymptote is y = 0. This occurs because as x becomes very large (positive or negative), the denominator grows much faster than the numerator, causing the fraction to approach zero.

2. Degree of Numerator = Degree of Denominator: If the degrees of the numerator and denominator are equal, the horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator). This means you divide the numbers in front of the highest power of x from the numerator and denominator. This is because, as x approaches infinity, the terms with the highest power of x will dominate the behavior of the function.

3. Degree of Numerator > Degree of Denominator: If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, there may be a slant (oblique) asymptote. In such cases, the function tends toward infinity (positive or negative) as x approaches infinity.

Formal Method (Limits):

The most rigorous method involves evaluating the following limits:

• lim_x→∞ f(x)
• lim_x→-∞ f(x)

If either of these limits exists and equals a finite number L, then y = L is a horizontal asymptote. It's possible to have different horizontal asymptotes as x approaches positive and negative infinity.

Important Considerations:

• Functions can cross their horizontal asymptotes. A horizontal asymptote describes the function's end behavior, not necessarily its behavior in the middle.
• For functions involving trigonometric functions or other functions that oscillate, the concept of a horizontal asymptote might need to be considered in a more nuanced way. The limits as x approaches infinity may not exist in the traditional sense, but the function's values may still be bounded within a certain range.