When does a rational function pass through the horizontal asymptote/how do I know how it behaves?

Question

Sorry if this is a bit simple compared to everything here, but I can't really seem to find an answer.

If I have $$f(x) = \frac{(x-2)(x-4)}{x(x-1)}$$ 1) When is the horizontal asymptote is crossed? Apparently to check if/where the horizontal asymptote is crossed I solve for f(x) = A, where A is the limit, is this true?

2)After solving for the vertical asymptotes I get x = 0 and x = 1. How do I know how each part behaves? My textbook made us use the behavior of the function as it got closer to the x intercepts, but that was for polynomial functions.

Answer 1

1) yes. $\lim{f(x)}_{x\to+\infty} = 1$ and $\lim{f(x)}_{x\to-\infty} = 1$

$f(x) = 1$ where $x = \frac{8}{5}$

2) after finding the vertical asymptotes look at the behavior of the function as you approach it from either side.

$\lim{f(x)}_{x\to0^-}= \infty$ and $\lim{f(x)}_{x\to0^+}= -\infty$

and $\lim{f(x)}_{x\to1^-}= -\infty$ and $\lim{f(x)}_{x\to1^+}= \infty$

You can determine those limits by looking at the graph, or by plugging in carefully chosen values of x. e.g. to determine $\lim{f(x)}_{x\to0^-}= \infty$ and $\lim{f(x)}_{x\to0^+}= -\infty$ notice that when $x=0$ the numerator is positive. Now pick two numbers on either side of 0, say $\pm\frac{1}{2}$. $-\frac{1}{2}$ gives a positive denominator, indicating $f(x)$ will take on larger and larger positive values. $\frac{1}{2}$ gives a negative denominator, indicating larger and larger negative values.

Answer 2

1) If either the $\lim_{x\rightarrow -\infty}f(x)=c_1$ or $\lim_{x\rightarrow +\infty}f(x)=c_2$ exists then you can get the horizontal asymptotes by evaluating $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$

If you let $y = f(x)$ then you will see the horizontal asymptotes will occur at the lines $y = c_1$ or $y = c_2$ (provided of course that those limits that defined $c_1,c_2$ exist)

Say you have a function with a horizontal asymptote that you found from evaluating the limit, it's possible (but not guaranteed) that the function has an intercept with the line defined by $y=c_1$. If this is the case (which it is for the f(x) in your question) then you can solve the simultaneous equations defined by $y = f(x)$ and $y=c_1$ or alternatively $f(x) = c_1$ to find the point on the graph that intercepts the horizontal asymptote line. (do the same for $c_2$ if it exists)

2) In a hand wavy way I think the easiest thing to do is to look at what happens on either side of the asymptote.