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5. The limit of a quotient is the quotient of the limits (provided
the limit of the denominator is not 0).
These laws are quite similar to the respective laws of sequences
stated earlier. Here are some more statements and examples about
functions.
Definition 12
We write lim
x→a
−
f(x)=L or lim
x→a−0
f(x)=L or f(a − 0) = L
and say the left-hand limit of f(x) as x approaches a (or the limit of
f(x) as x approaches a from the left)isequaltoL if we can make
values of f(x) as close to L as we like by taking x to be sufficiently
close to a and x less than a.
The symbol “x → a
−
” indicates that we consider only values of
x that are less than a. Similarly, if we require that x be greater than
a,weget“theright-hand limit of f(x) as x approaches a is equal to
L” and we write
lim
x→a+
f(x)=L or lim
x→a+0
f(x)=L or f(a +0)=L.
All main properties hold for one-sided limits, too.
The following is a true statement (it might be an effective tool
when discussing a limit of a given function): lim
x→a
f(x)=L if and
only if lim
x→a−0
f(x)=L and lim
x→a+0
f(x)=L.
Theorem 4
If f(x) g(x) when x is near a (except possibly at a)andthe
limits of f(x) and g(x) both exist as x approaches a,then
lim
x→a
f(x) lim
x→a
g(x).
Theorem 5 (The Squeeze Theorem)
If f(x) g(x) h(x) when x is near a (except possibly at a)
and lim
x→a
f(x) = lim
x→a
h(x)=L then lim
x→a
g(x)=L.
10
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