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L’Hôpital’s Rule

L’Hôpital’s Rule. 0/0 limits /  limits Growth/decay rates of exponentials, logs  -  limits 0  , 1  , 0 0 limits. Special Case. 0/0 L’Hôpital’s Rule. Suppose that lim f(x)=0 and lim g(x)=0 . Then. Examples of 0/0 limits. Compute the following limits. Warning and Strategy.

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L’Hôpital’s Rule

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  1. L’Hôpital’s Rule • 0/0 limits • /  limits • Growth/decay rates of exponentials, logs •  -  limits • 0, 1, 00 limits

  2. Special Case

  3. 0/0 L’Hôpital’s Rule • Suppose that lim f(x)=0 and lim g(x)=0. Then

  4. Examples of 0/0 limits • Compute the following limits

  5. Warning and Strategy • 4=limx1[(x2+x-6)/(x-2)]  limx1 (2x+1)/1=3 • Strategy for L’Hôpital: • Check that lim f(x)/g(x) is an indeterminate form. If not, cannot use L’Hôpital’s rule. • Differentiate f and g separately. • Find lim f (x)/ g (x). If limit is finite, +  or -  then it equals original limit, lim f(x)/g(x)

  6. /  limits • Suppose that lim f(x)=  and lim g(x)= . Then

  7. /  examples • Compute the following limits

  8. Growth/Decay rates of exponentials and logs

  9. Other indeterminate forms •  -  Example: limx0 (1/x - 1/sin x) • 1 Example: limx 0(1+x)1/x • 00Example: limx 0+ xx

  10. More Examples

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