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!!
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!