# L15 - Lecture 15 Indeterminate Forms; LHopitals Rule...

This preview shows pages 1–3. Sign up to view the full content.

Lecture 15 — Indeterminate Forms; L’Hˆopital’s Rule Indeterminate forms are limits for which we cannot plug in numbers immediately to evaluate nor can we reason the answer by inspection. These are NOT indeterminate forms: lim x →∞ arctan( x ) 1 + e - x lim x 0 x ln( x ) lim x →∞ 1 1 + x 2 lim x 1 1 1 - x 2 The indeterminate forms of interest to us are: ± 0 0 ² ± ² ( 0 · ∞ ) ( ) ( 1 ) ( 0 0 ) ( 0 ) For example, when looking at the quotient f ( x ) g ( x ) , suppose that when we try to use limit laws, we see that both f ( x ) 0 and g ( x ) 0 . The symbol 0 0 is not a real number, so the limit laws fail. In addition, we can not use the informal deﬁnitions to divine what will happen: if f ( x ) and g ( x ) are two numbers close to zero, their ratio could be anything! Thus, we need to manipulate the function by some other means in order to calculate the limit. Although we’ve seen many such techniques, we now learn an especially useful one called L’Hˆopital’s Rule.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
L’Hˆopital’s Rule Suppose that f and g are differentiable at a and g 0 ( x ) 6 = 0 in an open interval containing a (except possibly at a itself). Suppose that lim x a f ( x ) = 0 and lim x a g ( x ) = 0 or lim x a f ( x ) = ±∞ and lim x a g ( x ) = ±∞ Then lim x a f ( x ) g ( x ) = PROVIDED the second limit exists or is ±∞ . Thus, we have a new strategy for dealing with the
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 05/27/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

### Page1 / 11

L15 - Lecture 15 Indeterminate Forms; LHopitals Rule...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online