how do you find the asymptotes of a curve like:

\(\displaystyle (x+3)/(x^2+4)\)

where the denominator is a power for eg.

and why do you integrate \(\displaystyle 3/x\) to get 3lnx and not \(\displaystyle -9x^-3\)??

help very appreciated thanks!!

To get the asymptotes, test its end behaviors by taking both limits as \(\displaystyle x\rightarrow \infty\) and as \(\displaystyle x\rightarrow -\infty\). Both of which are 0, therefore there is only one asymptote which is \(\displaystyle y = 0\).

To answer your second question, the reason it turns into \(\displaystyle y = ln |x| \) is because when you are integrating using the reverse power rule which says

\(\displaystyle \int {x}^{r}dx = \frac{{x}^{r+1}}{r+1} \)

however this only works if \(\displaystyle r \neq -1\), because then denominator becomes 0. In your case, we have \(\displaystyle y = 3{x}^{-1}\), so \(\displaystyle r = -1\) so it doesn't work. So we use \(\displaystyle y = ln |x| \). Make sure that it's absolute value of x and not just regular x. If you don't put the absolute value signs, you'll get it wrong.

Cheers!