James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

appendix F Proofs of Theorems A45
Now we add f sad to both sides of this inequality:
f sad 1 f 9sadsx 2 ad , f sad 1 f 9scdsx 2 ad
But from Equation 1 we have f sxd − f sad 1 f 9scdsx 2 ad. So this inequality becomes
3 f sxd . f sad 1 f 9sadsx 2 ad
which is what we wanted to prove.
For the case where x , a we have f 9scd , f 9sad, but multiplication by the negative
number x 2 a reverses the inequality, so we get (2) and (3) as before.
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Section 4.4
In order to give the promised proof of l’Hospital’s Rule, we first need a generalization of
the Mean Value Theorem. The following theorem is named after another French mathematician,
Augustin-Louis Cauchy (1789–1857).
See the biographical sketch of Cauchy
on page 109.
1 Cauchy’s Mean Value Theorem Suppose that the functions f and t are continuous
on fa, bg and differentiable on sa, bd, and t9sxd ± 0 for all x in sa, bd. Then
there is a number c in sa, bd such that
f 9scd
t9scd
−
f sbd 2 f sad
tsbd 2 tsad
Notice that if we take the special case in which tsxd − x, then t9scd − 1 and Theorem
1 is just the ordinary Mean Value Theorem. Furthermore, Theorem 1 can be proved
in a sim ilar manner. You can verify that all we have to do is change the function h given
by Equation 4.2.4 to the function
hsxd − f sxd 2 f sad 2
and apply Rolle’s Theorem as before.
f sbd 2 f sad
tsbd 2 tsad
ftsxd 2 tsadg
L’Hospital’s Rule Suppose f and t are differentiable and t9sxd ± 0 on an open
interval I that contains a (except possibly at a). Suppose that
lim f sxd − 0 and lim tsxd − 0
x l a x l a
or that
lim f sxd − 6` and lim tsxd − 6`
x l a x l a
(In other words, we have an indeterminate form of type 0 0 or `y`.) Then
lim
x l a
f sxd
tsxd − lim
x l a
f 9sxd
t9sxd
if the limit on the right side exists (or is ` or 2`).
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