graph the function. label all intercepts and asymptotes.
y = x-2 / (x+1)(x-3)
i got for the intercepts: y= 2/3
x= -2
y-asymptote = 2
how will this graph look?
$\displaystyle \lim_{\text{ }x\to \mp \infty } \frac{x-2}{(x+1)(x-3)}$
$\displaystyle \lim_{\text{ }x\to \mp \infty } \frac{x-2}{x^2 - 2x -3}$
To calculate this limit, we divide both numerator and denominator by $\displaystyle x^n$, n is the highest power of x in the denominator. In $\displaystyle x^2 - 2x -3$, the highest power of x is 2. So n = 2. We'll divide both numerator and denominator by $\displaystyle x^2$.
$\displaystyle \lim_{\text{ }x\to \mp \infty } \frac{\frac{x-2}{x^2}}{\frac{x^2 - 2x -3}{x^2}}$
$\displaystyle \lim_{\text{ }x\to \mp \infty } \frac{\frac{1}{x}-\frac{2}{x^2}}{1 - \frac{2}{x} - \frac{3}{x^2}}$
When x goes to $\displaystyle \infty$, the fractions $\displaystyle \frac{1}{x}$, $\displaystyle \frac{1}{x^2}$, $\displaystyle \frac{2}{x}$, $\displaystyle \frac{3}{x^2}$... etc will approach 0.
So, the limit becomes $\displaystyle \frac{0}{1} = 0$. That means the horizontal asymptote is the line y=0, which is also called the x axis.