Question

How can I do this with the function (x+2)/x-3

Answer 1

(x+2)/(x-3) is already a rational function with a linear binomial in both the numerator and denominator...

The vertical asymptote occurs when the slope is undefined, in this case when you have division by zero, when x=3.  So the vertical asymptote is about the vertical line x=3.

The horizontal asymptote occurs about the horizontal line about the line y=k which f(x) approaches, but does not equal as x approaches ±oo.  Which in this case is relatively intuitive, but if you weren't sure, you could divide all terms by the highest power of x, in this case:

(x/x+2/x)/(x/x-3/x)  as x approaches oo is

(1+0)/(1-0)=1, so the horizontal asymptote is about the horizontal line y=1

The y-intercept occurs when x=0 so:

f(0) is just 2/-3=-2/3, so the y-intercept is the point (0, -2/3)