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Theorem 10 (Lagrange’s Theorem)
If f(x) is continuos on [a, b] and it is differentiable in (a, b), then
there is at least one c inside (a, b) with the property
f(b) − f(a)=f
(c)(b −a).
Theorem 11 (Cauchy’s Theo rem)
Let f(x),g(x) be differentiable in (a, b) and they are continuous
on [a, b]. Assume that g
(x) is never zero for x from (a, b). Then there
exists c inside (a, b) with the property
f
(c)
g
(c)
=
f(b) − f(a)
g(b) −g(a)
.
5.2. Limits Involving Infinity
Definition 19
Let f(x) be a function defined on both sides of a, except possibly
at a.Thenlim
x→a
f(x)=∞ means that for every number M there exists
such δ>0 that f(x) >M whenever 0 < |x − a| <δ.
Notation 2
Typically, we write just ∞ for +∞;thenotionlim
x→a
f(x)=−∞
is defined similarly to the just defined lim
x→a
f(x)=+∞.
5.3. Indeterminate Forms and L’Hospital’s Rule
Suppose we are trying to analyze the behavior of the function h(x)=
ln x
x − 1
. Although h is not defined when x =1, we need to know how
h behaves near 1. In particular, we would like to know the value of
the limit
lim
x→1
ln x
x −1
. (3)
But we can’t apply the respective Law of limits (the limit of a
quotient is the quotient of the limits) to (3) because the limit of the
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