Asymptotes and graphical features question

Question

Equation: y=((x+5)/(x+1)-:((x+3)(x-2))/((x-4)(x+1)))-1/(x-2)

a. Graph the original (not reduced) equation. Where are the vertical asymptotes located? Explain how you can use the equation to verify these vertical asymptotes.

1. I got the vertical asymptotes as x= -3, 2. I am not sure how to verify them using the equation.

b. What graphical feature occurs at x= -1? Why isn't there a vertical asymptote at x= -1?

Answer 1

Graphically I see an asymptotes at x=-1,2,4

From the equation y = (x + 5) / (x + 1) - ((x + 3) (x - 2)) / ((x - 4) (x + 1)) - 1 / (x - 2)

We know vertical asympotes occur where the denominator is zero. So x + 1 = 0 when x=-1. Remember that when x gets close to -1 the fraction (x + 5) / (x + 1) becomes large.

Also ((x - 4) (x + 1)) = 0 when either (x - 4)= 0 or (x + 1)=0. So when x=4 or -1.

(x - 2) = 0 when x=2.

For part b. there is a vertical asymptote at x=-1. There may be a typo in your equation.