VIDEO: Horizontal asymptotes simply explained

That is a horizontal asymptote

If you are in a task from the mathematics If you want to investigate a function and determine the horizontal asymptote, you should first know what that actually means.

• Should you have to determine a horizontal asymptote, that means that you should find a straight line that the given function approximates without touching it.
• Since it is supposed to be a horizontal asymptote, this means that the asymptote or the straight line sought should have a horizontal course, that is, the x-axis itself is or runs parallel to the x-axis.
• From a mathematical point of view, the function for large x-values ​​approaches this horizontal one Straight linesbut without reaching you.

How to determine the horizontal asymptote

• Horizontal asymptotes appear particularly frequently in the case of fractional rational ones Functions in which both the numerator and the denominator contain the variable x and possibly Potencies emerge from it. An example is the function f (x) = 1-x (x². But exponential functions or logarithmic functions can also have horizontal asymptotes.
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Determine asymptote

The request to determine the asymptote does not have to panic in anyone. …

• In order to determine a horizontal asymptote, one has to determine which limit value the function values ​​(y) strive towards when the x-values ​​go into positive infinite and negative infinite.
• Simplified, there is an infinitely large positive or negative value for the x-values. negative number and then see what happens to the function values.
• To do this, you proceed that you only consider the x-values ​​in the numerator and denominator with the highest power, since the other values ​​in infinity are neglected. If you have an x-value with any power in both the denominator and the numerator, you have to shorten the fraction and see whether and which number comes out.
• This number then describes where the horizontal asymptote of the function is so that you can easily draw it into your coordinate system.
• For the above example f (x) = 1-x / x² you get the x-axis as a horizontal asymptote, since the function values ​​for large x are arbitrarily small, that is, they approach zero. With the function f (x) = (2x²-1) / x² you get x = 2 as a horizontal asymptote if you follow the above rules (observe powers).

Note that not every fractional rational function has a horizontal asymptote. An example is the function x² / (1-x), which increases over all limits for large x.