
3-16 Calculus – Differentiation and Integration
Now, let is true for lim ( ) lim ( )
x a
n
x a
n
n
f x f x K n
→ →
[ ]
=
= −11
1
1
.
i.e.,
Consider,
lim ( ) lim ( ) .
l
x a
n
x a
n
n
f x f x K
→ →
−
−
[ ]
=
=
iim ( ) lim ( ) ( )
lim ( ) lim
x a
n
x a
n
x a
n
x a
f x f x f x
f x f
→ →
−
→
−
→
[ ]
=
=
1
1
(( )
.
x
K K
n n
=
−1
b. This is just a special case of (a).
i.e., lim ( ) lim ( )
lim ( )
/
/
x a
n
x a
n
x a
n
f x f x
f x
→ →
→
=
[ ]
=
1
1
==
→
lim ( ).
x a
n
f x
Hence, the theorem.
Note 3.3 As a result of the above properties, we shall have
lim ( ) ( )
x a
P x P a
→
=
for any polynomial function P x a x a x a x a
n n n
n
( ) .= + + + +
− −
0 1
1
2
2
In addition, we notice that the above definition ...