LIMIT OF A FUNCTION
Consider a function
f
(
x
) =
x
2
+
x
+ 1. As
x
approaches 1,
f
(
x
) gets closer to 3.
In mathematics, we write it as follows
lim
x
→
1
f
(
x
) = 3
(We say “the limit of
f
(
x
), as
x
approaches 1, equals 3.”)
We can see this also from the graph of the function, as we move the point
P
(
a, a
2
+
a
+ 1) on the graph closer to the point (1
,
3).
Suppose that we have a slightly different function,
f
(
x
) =
(
x
2
+
x
+ 1
if
x
6
= 1
2
if
x
= 1
.
Even though
f
(1) = 2, the point
P
(
a, a
2
+
a
+ 1) gets closer to the point (1
,
3) as
we move the point
P
. Thus, again, lim
x
→
1
f
(
x
) = 3.
Mathematically, the limit of a function is defined as follows:
“lim
x
→
a
f
(
x
) =
L
if we can make the values of
f
(
x
) arbitrarily close to
L
by taking
x
to be sufficiently close to
a
but not equal to
a
.”
We can see how it works through an example.
Example 1.
Guess the value of lim
x
→
1
x
2
+ 2
x

3
x
2

1
.
Solution
The function
f
(
x
) =
x
2
+ 2
x

3
x
2

1
is not defined when
x
= 1. However,
we only need to consider values of
x
not equal to 1. If we calculate
f
(
x
) for values
of
x
that approach 1, we can get the tables below.
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 Fall '09
 BenedictGross
 Math, Limit, Equals sign, lim h

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