The Real And Complex Number Systems

which implies that
Hence,
g 1 h g 1 0g 2 h g 1 hg 2 h g 2 0
fa h 2fa fa h
h 2
f a h f a h
2h
fa h 2fa fa h
lim
h0 h 2
f
lim
a h f a h
h0 2h
f a since f a exists.
(d) Give an example where the limit of the quotient in (c) exists but where f a does
not exist.
Solution: (STUDY) Note that in the proof of (c) by using L-Hospital Rule. We know
that |x| is not differentiable at x 0, and |x| satisfies that
|0 h| |0 h|
f
lim
0 lim
0 h f 0 h
h0 2h
h0 2h
So,letustrytofindafunctionf so that f x |x|. So, consider its integral, we know that
x 2
2
fx
if x 0
x2 if x 0
2
Hence, we complete it.
Remark: (i) There is a related statement; we write it as follows. Suppose that f defined
on a, b and has a derivative at c a, b. Ifx n a, c and y n c, b with such
that x n y n 0asn . Then we have
f fy
c lim n fx n
n y n x n
.
Proof: Sincef c exists, we have
fy n fc f cy n c oy n c *
and
fx n fc f cx n c ox n c. *’
If we combine (*) and (*’), we have
Note that
we have
fy n fx n
y n x n
y n c
y n x n
lim n
lim n
0.
which implies that, by (**)
f c oy n c
y n x n
1and
oy n c
y n x n
oy n c
y n c
x n c
y n x n
1 for all n,
ox n c
y n x n
y n c
y n x n
ox n c
x n c
ox n c
y n x n
. **
x n c
y n x n