l'Hôpital's Rule

Choose one of the following "forms" of limit, to which l'Hôpital's Rule can be applied:

Or, choose one of the following topics:

The limits of fractions of a certain type, such as

the limit as x goes to 1 of (ln(x))/(x-1),

cannot be found using the basic limit laws, since, in this case, the denominator's limit is zero, and there is no cancellation which can be done. Similarly,

the limit as x goes to infinity of (exp(x))/x

cannot be found directly either, since taking each limit separately results in infinity, and there aren't any arithmetic rules for infinity. However, l'Hôpital's Rule has been developed for just such cases.

l'Hôpital's Rule

Let f and g be differentiable functions, with g'(x) not zero in an interval around a, except possibly at a itself. Also, one of the following must hold true:
  • both f(x) and g(x) have limit 0 as x approaches a.
  • both f(x) and g(x) have infinte limit (either positive or negative) as x approaches a.

Then, the limit of the ratio f/g is equal to the limit of the ratio f'/g' (where the prime indicates the appropriate derivative), as long as that limit exists, or is infinite.

Choose one of the following "forms" of limit, to which l'Hôpital's Rule can be applied:

Or, choose one of the following topics:

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