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curve y = f(x) lies between the lines y = L − ε and y = L + ε.You
can see that if such a δ has been found, then any smaller δ will also
work. It is important to realize that such a process must work for
every positive number ε no matter how small it is chosen. Typically,
if a smaller ε is chosen, then a smaller δ may be required.
2.1. Calculating Limits Using the Limit Laws
We can use calculators and graphs to guess the values of limits,
but such methods don’t always lead to the correct answer. In this
paragraph we use properties of limits, called the Limit Laws,to
calculate limits.
2.2. Limit Laws
Suppose that c is a constant and the limits lim
x→a
f(x) and lim
x→a
g(x)
exist. Then
1) lim
x→a
[f(x)+g(x)] = lim
x→a
f(x) + lim
x→a
g(x),
2) lim
x→a
[f(x) − g(x)] = lim
x→a
f(x) − lim
x→a
g(x),
3) lim
x→a
[cf(x)] = c lim
x→a
f(x),
4) lim
x→a
[f(x)g(x)] = lim
x→a
f(x) lim
x→a
g(x),
5) lim
x→a
f(x)
g(x)
=
lim
x→a
f(x)
lim
x→a
g(x)
,iflim
x→a
g(x) =0.
These five laws can be stated verbally as follows:
1. The limit of a sum is the sum of the limits.
2. The limit of a difference is the difference of the limits.
3. The limit of a constant times a function is that constant times
the limit of the function.
4. The limit of a product is the product of the limits.
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