Function with two horizontal asymptotes and three vertical a

[blazerfan87]

I am having a lot of trouble with this question, give an example of a function that has two horizontal asymptotes and three vertical asymptotes. I know an example of a function having two horizontal asymptotes is y=tan^-1(x), and 1/(x^2)(x-1) has three vertical asymptotes, but when they are composed, it doesn't come out. any help on this topic would be greatly appreciated, Thanks.

 

[galactus]

Work backwards. Make up vertical asymptotes at x=1, x=-2, and x=3

You know that arctan has two HA. Try this one. I think it works:

$\displaystyle \frac{-3tan^{-1}(x)(x^{3}-4)}{5(x-1)(x+2)(x-3)}$

It has two HA. They are at $\displaystyle y=\pm\frac{3\pi}{10}$ and the VA are made up. Note the power of the numerator and denominator are the same.

Check it out and make sure. Test the limits as $\displaystyle x\to {\infty}$ and $\displaystyle x\to -\infty$

 

[daon]

Good one galactus,

Here's one using absolute value: $\displaystyle f(x) = |x-1|^3+(x-1)^3$. Let $\displaystyle a,b,c \neq 1$ with a,b,c all different.

Notice $\displaystyle \frac{f(x)}{(x-a)(x-b)(x-c)} \to 0$ as $\displaystyle x \to -\infty$

But $\displaystyle \frac{f(x)}{(x-a)(x-b)(x-c)} \to 2$ as $\displaystyle x \to \infty$