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Published byElisa Lakeman Modified over 2 years ago

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

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

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Using L’Hôpital’s Rule Twice

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

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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 0 0 First, compute the limit of the logarithm ln x x = x ln x: ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `` `

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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 n 2 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) = x 2 as x → ∞.whole number We say that f (x) grows faster than g (x) if ` `

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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 ondifferentiable provided that the limit on the right exists. A similar result holds for limits as x → ` ` ` ` `

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Which of f (x) = x 2 and g (x) = x ln x grows faster as x →

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

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