Download presentation

2
**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.

3
Note that the quotient is still indeterminate at x = π/2. We removed this indeterminacy by cancelling the factor − cos x.

4
**Using L’Hôpital’s Rule Twice**

5
Assumptions Matter Can L’Hôpital’s Rule be applied to `

6
**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: ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ``

7
**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 ` `

8
**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 →

9
**Which of f (x) = x2 and g (x) = x ln x grows faster as x →**

10
Jonathan is interested in comparing two computer algorithms whose average run times are approximately (ln n)2 and THEOREM 3

Similar presentations

Presentation is loading. Please wait....

OK

Asymptotic Growth Rate

Asymptotic Growth Rate

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on universe and solar system Ppt on itc group of hotels Ppt on indian defence services Ppt on any topic of science Ppt on therapeutic environment in nursing Short ppt on unemployment in india Ppt on acute and chronic diseases for class 9 Ppt on game theory five nights Ppt on global warming with pictures Ppt on forward rate agreement explain