3.1 Limits and Continuity
Example 1. Investigate the behavior of the function f defined by f (x) = x2 − 3x + 8 for values of x
near 1. The following table gives values of f (x) for values of x close to 1 but not equal to 1.
x
f (x)
0.0
8.000000
0.5
6.750000
0.8
6.240000
0.9
6.110000
0.95
6.052500
0.99
6.010100
0.995
6.005025
0.999
6.00100
x
f (x)
2.0
6.000000
1.5
5.750000
1.2
5.840000
1.1
5.910000
1.05
5.952500
1.01
5.990100
1.005
5.995025
1.001
5.999001
We write
and say “the limit of f (x), as x approaches a, equals L” if we can make the values of f (x) arbitrarily
close to L (as close to L as we like) by taking x to be sufficiently close to a but not equal to a.
In the above example:
NOTE: f (x) does not need to be defined when x = a. The only thing that matters is how f is defined
near a.
x−1
Example 2. Use the given table (correct to 6 decimal places) to estimate lim 3
numerically.
x→1 x − 1
x−1
Notice that 3
is undefined at x = 1.
x −1
x
0.2
0.6
0.8
0.9
0.99
0.999
1.8
1.4
1.2
1.1
1.01
1.001
f (x)
x
f (x)
1
We write
and say the
(or the
) is equal to L if we can
make the values of f (x) arbitrarily close to L by taking x to be sufficiently close to a and x less than
a.
We write
and say the
(or the
) is equal to L if we can
make the values of f (x) arbitrarily close to L by taking x to be sufficiently close to a and x greater
than a.
The following is true:
2
Example 3. (Stewart) For the function f whose graph is given, state the value of the given quantity,
if it exists.
a) lim f (x)
d) lim f (x)
g)
b) lim− f (x)
e) f (3)
h) lim f (x)
c) lim+ f (x)
f)
x→1
x→3
x→3
x→3
lim f (x)
x→−2+
x→−2
lim f (x)
i) f (−2)
x→−2−
Let f be a function defined on both sides of a, except possibly at a itself. Then lim f (x) = ∞ means
x→a
that the values of f (x) can be made arbitrarily large (as large as we please) by taking x sufficiently
close to a (but not equal to a).
Let f be defined on both sides of a, except possibly at a itself. Then lim f (x) = −∞ means that the
x→a
values of f (x) can be made arbitrarily large negative by taking x sufficiently close to a (but not equal
to a).
(Image from Stewart.)
The line x = a is called a
the following statements is true:
of the curve y = f (x) if at least one of
3
Example 4. Use the given table (correct to 6 decimal places) to estimate lim+
x→2
cally.
x
3
2.1
2.01
2.001
f (x)
Limit Laws: Suppose that c is a constant and the limits
lim f (x) and lim g(x)
x→a
x→a
exist. Then
1. lim [f (x) ± g(x)] =
x→a
2. lim [cf (x)] =
x→a
3. lim [f (x)g(x)] =
x→a
f (x)
=
x→a g(x)
4. lim
5. lim [f (x)]n =
x→a
6. lim c =
x→a
7. lim xn =
x→a
8. lim
√
n
x=
9. lim
p
n
f (x) =
x→a
x→a
4
1 − x2
numerix2 + 3x − 10
2.0001
2.00001
Example 5. Evaluate the following limits using the Limit Laws.
a) lim (5x2 − 2x + 3)
x→4
b) lim (x2 + 1)(x2 + 4x)
x→2
c) lim
x→−1
d) lim
t→−1
x2
x−2
+ 4x − 3
√
t3 + 2t + 7
e) lim(r4 − 7r + 4)2/3
r→3
5
If f is a polynomial or a rational function and a is in the domain of f , then
Example 6. Evaluate each limit, if it exists.
x2 − x − 12
x→−3
x+3
a) lim
x2 − x − 2
x→−1
x+1
b) lim
(h − 5)2 − 25
h→0
h
c) lim
9−t
√
t→9 3 −
t
d) lim
e) lim
x→0
1
x
6
Example 7. Let
f (x) =
x2 − 2x + 2 if x < 1
3−x
if x ≥ 1
a) Find lim− f (x) and lim+ f (x).
x→1
x→1
b) Does lim f (x) exist?
x→1
c) Sketch the graph of f .
7
Continuity:
A function f is
if
1.
2.
3.
Geometrically, you can think of a function that is continuous at every number in an interval as a
function whose graph has no break in it. The graph can be drawn without removing your pencil from
the paper.
A function f is continuous on an interval if it is continuous at every number in the interval.
Any polynomial is continuous everywhere.
Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.
Example 8. From the graph of f ,
a) State the intervals on which f is continuous.
b) State the numbers at which f is discontinuous and state which condition of the definition of continuity
is broken.
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Example 9. Explain why each function is discontinuous at the given point.
x2 − 1
at a = −1
a) f (x) =
x+1

 x2 − 2x − 8
x 6= 4
b) f (x) =
at a = 4
x
−
4
 3
x=4
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Example 10. Let

 x x<0
x2 0 ≤ x ≤ 1
g(x) =
 3
x x>1
Show that g(x) is continuous on (−∞, ∞).
Example 11. Find the points at which f is discontinuous.
 √
 −x x < 0
1
0≤x≤1
f (x) =
 √
x
x>1
Example 12. Find the value of the constant c that makes h continuous on (−∞, ∞).
2
x − c2 x < 4
h(x) =
cx − 9 x ≥ 4
10