Question

What are the asymptotes for `((x+2)(x+5))/(x+2)`?

Answer 1

There are no vertical or horizontal asymptotes

Explanation:

`y=((x+2)(x+5))/(x+2)`

Step 1. Cross out any factors in both the numerator and denominator.

`y=color(red)(cancel(color(black)("("x+2")"))color(black)("("x+5")"))/color(red)cancel(color(black)(x+2)`

This reduces the function to the line `y=x+5` with a removable discontinuity (a little hole in the graph) where `x=-2` and `y=-2+5=3`.

So the hole is at (-2,3).

Step 2. Find the vertical asymptotes.

Since the denominator is now 1, there are no vertical asymptotes.

Step 3. Find the horizontal asymptotes.

This is a linear equation. There are no horizontal asymptotes.

There are no horizontal or vertical asymptotes.