Math Help - Solving Rational Equation's Oblique(slant) Asymptote

1. Solving Rational Equation's Oblique(slant) Asymptote

I have a quick question, the oblique asymptote of this rational equation is as follows:

f(x)=(x^2-2x-5)/(x-1)

oblique asymptote: y=x-1

But my question is, would you write the answer as
y=x-1
or would you write it like you are answering for a horizontal asymptote ie.
as x -> +-infinity, y -> 1

and one more question, is it true that a function CAN cross the horizontal asymptote but cannot cross the vertical one?

thanks

2. Asymptotes

Hello skeske1234
Originally Posted by skeske1234
I have a quick question, the oblique asymptote of this rational equation is as follows:

f(x)=(x^2-2x-5)/(x-1)

oblique asymptote: y=x-1

But my question is, would you write the answer as
y=x-1
or would you write it like you are answering for a horizontal asymptote ie.
as x -> +-infinity, y -> 1

and one more question, is it true that a function CAN cross the horizontal asymptote but cannot cross the vertical one?

thanks
I think both ways make sense: it just depends what the question is. If it is 'What is the equation of the asymptote?' then the answer is $y=x-1$. (Or, in the case of the horizontal asymptote you mention: $y=1$).

But if the question is: 'Describe the behaviour of the function as $x \rightarrow\infty$' then you could say things like 'As $x \rightarrow \infty, y \rightarrow x-1$'

As to your second question, since a function has at most only one value for each value of $x$, a function cannot cross any vertical line, whether it's an asymptote or not.