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Mathematics
Problem
In the course of proving the Constant Multiple Rule, a student writes
F
′
(
x
)
=
lim
h
→
0
c
f
(
x
+
h
)
−
c
f
(
x
)
h
(1)
F'(x)=\displaystyle\lim_{h\to0}\frac{c\,f(x+h)-c\,f(x)}{h}\quad\text{(1)}
F
′
(
x
)
=
h
→
0
lim
h
c
f
(
x
+
h
)
−
c
f
(
x
)
(1)
=
c
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
(2)
=\displaystyle c\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\quad\text{(2)}
=
c
h
→
0
lim
h
f
(
x
+
h
)
−
f
(
x
)
(2)
Which property of limits justifies the transition from step (1) to step (2)? State this property.
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