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For the conditions we have established 0
k h .Soif h
, then
certainly k
.
What this establishes is a convenient rule for finding diMcult limits. If
f ( a )
g ( a )
0, then our ordinary procedures for finding lim
f ( x )/ g ( x )
may not beused, but wecan try to find lim
f
( x )/ g
( x ), and, if weare
successful, that is the answer we are looking for.
27 (i) 1/
2. (ii) 1. (iii) 1, 1/2, 1/6. The first result in (iii) is of great importance
and is variously justified in different treatments. Some make it plausible
from geometric considerations and then use it to find the derivative of sine.
In such a case, de l'Hoˆ pital's rulemay not beused, for that would makea
circular argument. Some treatments define sine by a power series, and then
del'Hoˆ pital's rule is needed for a sound argument. We will define sine
analytically (but with geometric motivation) in chapter 11, and that
provides a proper basis for the argument here.
28 The basis of the question is the application of Cauchy's Mean Value
Theorem in case f ( a ) g ( a ) 0. In this case, for b a , wehave
f ( b )
g ( b )
f ( c )
g ( c ) for some c , with a c b .
If, given B , there exists a
such that
f
( x )
a x a
( x ) B ,
g
f ( y )
g ( y )
f
( x )
g
then a y a
( x ) B
for some x , with a x y a . The result is proved.
29 After applying de l'Hoˆ pital's rule, use qn 8.23.
f ( x ) 2 x for x 0, and f ( x ) 2 x for x 0. This f is not differentiable
at 0. But the limit exists and equals 0.
30 For the first result see qn 8.21.
f
( x )/ g
cos(1/ x ), of which thefirst component tends to 0
as in qn 8.21, but the second component oscillates near 0 like the function
of qn 8.20.
Cauchy's Mean Value Theorem says that, if a
( x )
2 x sin(1/ x )
b , then there exists a c
with a
a there is nothing to indicate that
the consequent c s for each b exhaust the possibilities.
c
b , such that . . .. But if b
31 F ( a ) 0 from thedefinition of K . F ( b ) 0, trivially. F is differentiable from
qns 8.10 and 8.13.
32 f ( a h )
f ( a )
hf
( a )
h f
( a h ), for some
,0
1. Put a
0
and h x .
 
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