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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_06” 2011/5/22 23:14 page 22 #22
6-22 Real Analysis
We now use the definition of derivative and the properties of
the exponential function (Theorem 5.7.4) to compute the derivative
of f (x) = e
x
. Indeed,
f
(x) = lim
h0
f (x + h) f (x)
h
= lim
h0
e
x+h
e
x
h
= e
x
lim
h0
e
h
1
h
= e
x
.
Using Theorem 6.2.12, we can also compute the derivative of g(y) =
log y. Indeed, using this theorem
g
(y) =
1
f
(x)
=
1
f (x)
=
1
y
,
for all y > 0 where f (x) = y or g(y) = x.
6.5 L’HOSPITAL’S RULE
Using theorems on limits of functions, we see that if f (x) α and
g(x) β as x a where α, β are extended real numbers, a
[−∞, ∞], then
f (x)
g(x )
α
β
as x a whenever
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Publisher Resources

ISBN: 9781299447561Publisher Website