Indeterminate Form - Evaluating Indeterminate Forms

Evaluating Indeterminate Forms

The indeterminate nature of a limit's form does not imply that the limit does not exist, as many of the examples above show. In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated.

For example, the expression x2/x can be simplified to x at any point other than x = 0. Thus, the limit of this expression as x approaches 0 (which depends only on points near 0, not at x = 0 itself) is the limit of x, which is 0. Most of the other examples above can also be evaluated using algebraic simplification.

L'Hôpital's rule is a general method for evaluating the indeterminate forms 0/0 and ∞/∞. This rule states that (under appropriate conditions)

where f' and g' are the derivatives of f and g. (Note that this rule does not apply to forms like ∞/0, 1/0, and so on; but these forms are not indeterminate either.) With luck, these derivatives will allow one to perform algebraic simplification and eventually evaluate the limit.

L'Hôpital's rule can also be applied to other indeterminate forms, using first an appropriate algebraic transformation. For example, to evaluate the form 00:

The right-hand side is of the form ∞/∞, so L'Hôpital's rule applies to it. Notice that this equation is valid (as long as the right-hand side is defined) because the natural logarithm (ln) is a continuous function; it's irrelevant how well-behaved f and g may (or may not) be as long as f is asymptotically positive.

Although L'Hôpital's rule applies both to 0/0 and to ∞/∞, one of these may be better than the other in a particular case (because of the possibilities for algebraic simplification afterwards). You can change between these forms, if necessary, by transforming f/g to (1/g)/(1/f).