When you are given a certain problem of solving limit function, I would suggest you to try following methods. Almost everybody would be faimiliar with the first method (just plugging into numbers directly into the expression). If it work, be happy... but if it does not work, get frustrated and give up. However, there are many different ways you can use to find the solution for Limit expression. Of course, at the first trial I would suggest you to try using 'plug in number' method because it is the simplest. However, if it does not work, don't get panic and try other methods one by one.

• Which Methods Can You Try?
• How Does Each Method Work on an Example?
• In Which Order Should You Try Them?

Which Methods Can You Try?

Every method on this page answers the same question: what value does f(x)/g(x) approach as x approaches a? They differ in what they do to the expression before the final plug in. If limits themselves are new to you, read the Limit page first.

The diagram below starts from the limit on the left and fans out into five methods. Each method ends with the same check: do you get a meaningful result?

• Plug in number : substitute x = a directly. This is the first thing to try, because it is the only method that needs no preparation.
• Apply L'Hospital's Rule and then plug in : replace f and g by their derivatives. It applies when plugging in gives 0/0 or ∞/∞.
• Factor it and then plug in : cancel a factor that makes both f and g zero at a.
• Multiply top and bottom with the conjugate and then plug in : remove a square root that causes the 0/0.
• Plot the formula and see where it approaches : estimate the limit from a graph or a table when no algebra suggests itself.

A meaningful result is an ordinary number. The results that are not meaningful are the forms 0/0 and ∞/∞, which say nothing yet about the limit, and a nonzero number over 0, which says the function grows without bound near a.

How Does Each Method Work on an Example?

Let's run each method on one small example, so that you can see what it changes in the expression. Every result below was checked with a computer algebra system and by evaluating the expression close to the point.

Plug in Number

Plugging in works whenever the function is continuous at the point. Take lim_x→1 (x² + 1)/(x + 1). Plugging in x = 1 gives 2/2 = 1, which is a meaningful result, so the limit is 1. The method fails only when the result is not a number.

L'Hospital's Rule

When plugging in gives 0/0 or ∞/∞, L'Hospital's Rule replaces f and g by their derivatives. For lim_x→0 (sin x)/x, plugging in gives 0/0. The derivatives are cos x and 1, and plugging in now gives cos 0/1 = 1. Both f and g must be differentiable near a, and the rule applies to those two forms only. The L'Hospital's Rule page covers the conditions in detail.

Factor It

Factoring works when the 0/0 comes from a factor that f and g share. For lim_x→2 (x² - 4)/(x - 2), the numerator factors as (x - 2)(x + 2). Cancelling x - 2 leaves x + 2, and plugging in gives 4. The cancellation is allowed because the limit never uses x = 2 itself.

Multiply by the Conjugate

This method targets a square root that causes the 0/0. For lim_x→0 (√(x + 4) - 2)/x, plugging in gives 0/0. Multiply the top and the bottom by the conjugate √(x + 4) + 2. The top becomes (x + 4) - 4 = x, which cancels the bottom, and the expression becomes 1/(√(x + 4) + 2). Plugging in now gives 1/4.

Plot the Formula

A plot or a table is the fallback when no algebra suggests itself. It gives an estimate, not a proof. For lim_x→0 (1 - cos x)/x², the values in the table below head for 0.5. L'Hospital's Rule applied twice confirms the exact value, 1/2.

In Which Order Should You Try Them?

The diagram lists the methods as parallel branches, and it leaves the order open. In practice the formula itself usually points to the right branch. The bullets below give one working order and the clue for each step.

• Plug in first, always : if it gives a number and the function is continuous there, you are done.
• Read the form of the failure : 0/0 and ∞/∞ can still have a limit, and the other methods exist for them. A nonzero number over 0 means the function grows without bound, so check the two one-sided limits.
• Match the method to the formula : polynomials suggest factoring, square roots suggest the conjugate, and sin x, e^x or log x suggest L'Hospital's Rule.
• Check the answer with a plot : a quick table near the point catches sign errors and wrong cancellations.

Let's walk one problem through that order: lim_x→9 (√x - 3)/(x - 9). Plugging in x = 9 gives 0/0, so the first step fails in the way the other methods can handle. The formula holds a square root, which points to the conjugate. Multiplying the top and the bottom by √x + 3 turns the top into x - 9. That cancels the bottom and leaves 1/(√x + 3), so plugging in gives 1/6.

The other branches agree. L'Hospital's Rule gives (1/(2√x))/1, which is also 1/6 at x = 9. A table gives 0.16662 at x = 9.01 and 0.16671 at x = 8.99, both close to 1/6 = 0.16667. When two methods and a table agree, the answer is safe.

A nonzero number over 0 needs different handling. For 1/x at x = 0, plugging in gives 1/0. The right-side limit is +∞ and the left-side limit is -∞, so the two-sided limit does not exist. None of the five methods changes that, because the function really does grow without bound.