Have linear functions asymptotes

What are vertical, horizontal, and crooked asymptotes?

Horizontal asymptote

As you can see in the following picture, the function graph approaches the straight line $ y = 2 $. That is also the horizontal asymptote. One method of finding an equation for this level is to create a limit. This limit value, which we only specify here, is obtained via so-called zero sequences. For the limit value one writes as follows:

$ \ lim_ {x \ to \ infty} \ frac {2x + 10} {x-5} = 2 $

and reads: Limes of $ \ frac {2x + 10} {x - 5} $ for x towards infinity equals $ 2 $.

Mapping of the function $ f (x) = \ frac {2x + 10} {x-5} $ with the asymptote $ a (x) = 2 $

The function you see here also has a vertical asymptote at $ x = 5 $. This is because you cannot substitute 5 for x, because then the denominator is zero and this division is not explained.
The function in the previous section continues to approximate the $ x $ axis. This axis is a horizontal asymptote. You can see this in the picture above.

Skewed asymptotes

Here we see the function $ {f (x) = \ frac {x ^ 2 + 1} {x}} $

Illustration of crooked asymptote

As you can see, the function has a skewed asymptote. With a fractional-rational function, this is always the case when the degree of the numerator polynomial is one higher than the degree of the denominator polynomial. The numerator polynomial is quadratic (x is quadratic), the denominator polynomial is linear (x is linear).
A equation the skewed asymptote is obtained in this case by rewriting the fraction:
$ {\ frac {x ^ 2 + 1} {x}} = {\ frac {x ^ 2} {x}} + {\ frac {1} {x}} = x + {\ frac {1} {x} } $
The result of this transformation consists of a linear part $ (x) $ and a fraction $ {\ frac {1} {x}} $. The linear part represents a equation the crooked asymptote So it is: $ a (x) = x $.
The $ y $ axis is also an asymptote here, but a vertical one at $ x = 0 $. This is because you cannot substitute zero for $ x $, because this division is not explained.


For the asymptotes of a fractional rational function $ {f (x) = \ frac {u (x)} {v (x)}} $ the following applies:

There are

... a vertical asymptote at the point x, if the denominator for this x is zero, but the numerator is not.
... a horizontal asymptote if the numerator polynomial is at most equal in degree to that of the denominator polynomial.
... a skewed asymptote if the numerator polynomial is exactly one greater in degree than the degree of the denominator polynomial.

With the Exercises you can deepen your knowledge about asymptotes. We wish you a lot of fun and success!