A vertical asymptote is a dashed vertical line that as the graph approaches the number that makes the function undefined, it approaches this line and f(x) blows up to infinity (or - infinity). A vertical asymptote is y = a, where x is any point that makes the denominator 0. You may also have to check if it's a hole or not.
A horizontal asymptote is a dashed horizontal line that as the graph approaches, it extends to infinity in the x and approaches the horizontal asymptote. A horizontal asymptote is x = b. To find the horizontal asymptote, take the limit of a rational function as it goes to infinity.
Examples:
1.
When does the vertical asymptote exist? When denominator is 0. When does the denominator become 0? When x = 2. To find out what happens on both sides of the asymptote, take the limit as x goes to 2 from both sides.
2.
What are the vertical asymptotes here? x = 1 only. Why not x = -1? Because, as you can see (x+1) cancels, and this implies that there is a hole at x= -1, not an asymptote.
3.
Finding the horizontal asymptote is simply taking the limit as x approaches infinity. If limit is infinity, no horizontal asymptote.
Take the first derivative to find where it is increasing/decreasing, the second derivative to find where it is concave up/down.