L'Hospital's rule 4.26. Prove L'Hospital's rule for the case of the "indeterminate forms" (a) 0/0 and (b) 0/00. (a) We shall suppose that f(x) and g(x) are differentiable in a
L'Hospital's rule 4.26. Prove L'Hospital's rule for the case of the "indeterminate forms" (a) 0/0 and (b) 0/00. (a) We shall suppose that f(x) and g(x) are differentiable in a
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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4.26) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.
Transcribed Image Text:L'Hospital's rule
4.26.
Prove L'Hospital's rule for the case of the "indeterminate forms" (a) 0/0 and (b) ∞/0.
(a) We shall suppose that f(x) and g(x) are differentiable in a <x<b and f(x,) = 0, g(x,) = 0, where a<x,<b.
By Cauchy's generalized mean value theorem (Problem 4.25),
f(x) _ f(x)- f(x,) _ f'CE)
g(x)- g(x,) g'E)
X。くらくx
g(x)
Then
f(x)
lim
f'(x)
f'(E)
= lim
= lim
= L
x→Xo+ g(x)
x→X,+ g'(E)
g'(x)
since as x → xo+, Š → xo+.
Modification of this procedure can be used to establish the result if x → xo -, x → Xo, X → 0, or x→-0.
(b) We suppose that f(x) and g(x) are differentiable in a <x< b, and lim f(x) = ∞, lim g(x)=∞ where
x→x,+
x→x,+
a<xo<b.
Transcribed Image Text:Assume x, is such that a < xo <x<x¡ <b. By Cauchy's generalized mean value theorem,
f(x)- f(x,) _ f'()
x< < x,
%3D
g(x)- g(x,)
g'(5 )
Hence,
f(x)- f(x,) _ f(x) 1- f(x,)/ f(x) f'C)
g(x)- g(x,)
g(x) 1- g(x,)/g(x) g'(5)
from which we see that
f(x) _ f'(E) 1-g(x,)/g(x)
g(x) g'() 1-f(x,)/f(x)
(1)
f'(x)
x>Xo + g'(x)
Let us now suppose that lim
= L and write Equation (1) as
f(x)
( f'().
1- g(x, )/g(x)
1-8(x, Vg(x)
- L
+ L
(2)
8(x)
g'(x)
1- f(x, )/f(x)
We can choose x, so close to x, that |f'(5)/g'(E) - L| < e. Keeping x, fixed, we see that
1-8(x,)/g(x)
lim
|=1 since 1 lim f(x), = ∞ and lim g(x) = 0
1-f(x,)/f(x)
xXo +
xXo +
x-Xo +
Then taking the limit as x → xo+ on both sides of (2), we see that, as required,
f(x)
f'(x)
= L = lim
lim
x→Xq+ g(x)
x->Xo+ g'(x)
Appropriate modifications of this procedure establish the result if x → xo -, x → Xo, x →∞, or x → -.
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