L’Hôpital’s Rule states that when has an at x = a, then we can replace f(x)/g(x) by the quotient of the derivatives f (x)/g (x). indeterminate form of type 0/0 or THEOREM 1 L’Hôpital’s Rule Assume that f (x) and g (x) are differentiable on an open interval containing a and that f (a) = g (a) = 0 Also assume that (except possibly at a). Then This conclusion also holds if f (x) and g (x) are differentiable for x near (but not equal to) a and if the limit on the right exists or is infinite Furthermore, this rule if valid for one-sided limits.

Note that the quotient is still indeterminate at x = π/2. We removed this indeterminacy by cancelling the factor − cos x.

Using L’Hôpital’s Rule Twice

Assumptions Matter Can L’Hôpital’s Rule be applied to `

The Form 00 First, compute the limit of the logarithm ln xx = x ln x: Limits of functions of the form f (x)g (x) can lead to the indeterminate forms In such cases, take the logarithm of the expression and then apply L’Hôpital’s Rule. The Form 00 First, compute the limit of the logarithm ln xx = x ln x: ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ``

Comparing Growth of Functions Sometimes, we are interested in determining which of two functions, f (x) and g(x), grows faster. For example, there are two standard computer algorithms for sorting data (alphabetizing, ordering according to rank, etc.): Quick Sort and Bubble Sort. The average time required to sort a list of size n has order of magnitude nlnn for Quick Sort and n2 for Bubble Sort. Which algorithm is faster when the size n is large? Although n is a whole number, this problem amounts to comparing the growth of f (x) = x ln x and g (x) = x2 as x → ∞. We say that f (x) grows faster than g (x) if ` `

To indicate that f (x) grows faster than g (x), we use the notation ` ` ` ` ` THEOREM 2 L’Hôpital’s Rule for Limits at Infinity Assume that f (x) and g(x) are differentiable on provided that the limit on the right exists. A similar result holds for limits as x →

Which of f (x) = x2 and g (x) = x ln x grows faster as x →

Jonathan is interested in comparing two computer algorithms whose average run times are approximately (ln n)2 and THEOREM 3