State the horizontal asymptote of the rational function. f(x) = quantity x squared plus six x minus eight divided by quantity x minus eight

There is no horizontal asymptote for the function f(x).

Step-by-step explanation:

We have the function . Notice that if the function is not defined, because the denominator of the fractions equals zero, and the numerator don't. This fact is equivalent to the existence of a vertical asymptote at . In mathematical language:

and

Now, in case that f(x) has an horizontal asymptote the following must hold:

But, actually

Hence, there is no horizontal asymptote.

Anyway, f(x) has an asymptote, but no horizontal. In order to obtain the slope of the asymptote, we need to find the following limit:

Then asymptote has equation . To find we calculate the limit

Hence, the asymptote at is .

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