# UCSC MATH 11A - 01 - L’Hospital’s rule (4 pages)

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## L’Hospital’s rule

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## L’Hospital’s rule

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Lecture Notes

Pages:
4
School:
University of California, Santa Cruz
Course:
Math 11a - 01 - Calculus With Apps
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143 4 8 L Hospital s rule limits revisited Textbook pages 299 307 L Hospital s rule is a very powerful tool for finding limits of indeterminate limits such as Rule for the indeterminate form 00 Examples for the indeterminate form 00 Find the limit of sin x x at x 0 Find the limit of x2 9 x 3 at x 3 Find the limit of tan x x at x 0 144 CHAPTER 4 DIFFERENTIAL CALCULUS Proof for the indeterminate form 00 Rule for the indeterminate form Examples for the indeterminate form Find limx ln x x Find limx 2 tan x 1 tan x The proof of this indeterminate form is more difficult so we will simply assume that it is true Advantages L Hospital Rule can be applied multiple times 4 8 L HOSPITAL S RULE LIMITS REVISITED 145 L Hospital Rule can also be used to find limits of other indeterminate forms provided we can cast them into either of the above 0 0 or Examples Find limx 0 x2 sin2 x Find limx 0 1 sin x cot x Important note L Hospital rule ONLY works for these two indeterminate forms Do not try to use it without verifying that the limit is one of those forms first For example you cannot use L Hospital Rule on the following problems limx 3 x2 1 x 3 limx 0 x ln x limx e x x2 and so forth 146 CHAPTER 4 DIFFERENTIAL CALCULUS Check your understanding of Lecture 22 Limits using L Hospital s Rule Do as many problems as you can from the following list Textbook page 307 numbers 1 through 32 a good start would be all of the odd numbered or all of the even numbered problems on that list Further examples of L Hospital s Rule Textbook page 308 number 51 56

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