The Indeterminate Form (0/0) L'HOPITAL'S RULE (0/0) Suppose that (11.5.1) fx)→0andg(x)→0 and in the approach g(x)and g'(x)are never 0. If f'(x) f(x) then g'(x) →y, g(x) →y Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Indeterminate Form (0/0)
The Indeterminate Form (0/0) Example Find lim- x0sin√ Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Indeterminate Form (0/0) Example Find 0 lim x sin x x → +
The Indeterminate Form (0/0) THEOREM 11.5.2 THE CAUCHY MEAN-VALUE THEOREM Suppose that f and g are differentiable on (a,b)and continuous on [a,b].If g'is never 0 in (a,b),then there is a number r in (a,b)for which f(r)f(b)-f(a) g'(r)g(b)-g(a) Main Meny cn8a
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. The Indeterminate Form (0/0)
Other Indeterminate Forms L'HOPITAL'S RULE (o/) Suppose that f(x)>±oo and g(x)→土00 (11.6.1) and in the approach g'(x)0. If 得为 then 8→ f(x) Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Other Indeterminate Forms
Other Indeterminate Forms Inx asx→o, xa →0 Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Other Indeterminate Forms
er asx→0o, →00 Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Other Indeterminate Forms Example Determine the behavior of 2”asn→o. a=- n Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Other Indeterminate Forms Example Determine the behavior of as n→∞. 2 2 n n a n =
Other Indeterminate Forms The Indeterminates0,1∞，ooo lim x*1. →0H Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Other Indeterminate Forms The Indeterminates 00 , 1 ∞, ∞0
Other Indeterminate Forms Example(1o） Find lim (1+x)s x→0 Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Other Indeterminate Forms Example (1∞) Find ( ) 1/ 0 lim 1 x x x → + +