These are the the general steps while you try to compute the graph of a function \(f\):

1. Determine domain of the \(f\)
2. Find \(f'(x)\) and use it to determine intervals on which \(f\) is increasing or decreasing, and local maximum or minimum points.
3. Find \(f''(x)\) and use it to determine intervals on which \(f\) is concave up or down, and inflection points.
4. Use limits to find all vertical and horizontal asymptotes.
5. Plot important points and sketch the curve!

Notes:

• If \(f(x)\) is undefined at some point, investigate what happens when \(x\) approaches the point from it's left and right side.
• If the limit of the above computation is infinity (with different signs) on different directions, they you have a vertical asymptotes.
• To check for horizontal asymptote, you can compute the limit of the function with \(x \to \infty\) and \(x \to -\infty\) and see if they give the same value.
• Wikipedia link for computing horizontal/vertical asymptotes

Suppose \(f\) is a function whose domains contains an interval \((a, \infty)\), and let \(A\) be the set of all the values of \(f\) at numbers in this interval; that is, \(A = { f(x): x > a} = {y: \text{for some } x > a, y = f(x)}\).

1. If \(f\) is weakly increasing on \((a, \infty)\) and \(A\) has an upper bound, then \(\lim_{x \to \infty} f(x)\) is defined and is equal to the least upper bound of \(A\).
2. If \(f\) is weakly increasing on \((a, \infty)\) and \(A\) does not have an upper bound, then \(\lim_{x \to \infty} f(x) = \infty\)
3. If \(f\) is weakly decreasing on \((a, \infty)\) and \(A\) has a lower bound, then \(\lim_{x \to \infty} f(x)\) is defined and is equal to the greatest lower bound of \(A\).
4. If \(f\) is weakly decreasing on \((a, \infty)\) and \(A\) does not have a lower bound, then \(\lim_{x \to \infty} f(x) = -\infty\)