3.1 Limits and Continuity
Example 1. Investigate the behavior of the function f defined by f (x) = x2 − x + 2 for values of x
near 2. The following table gives values of f (x) for values of x close to 2 but not equal to 2.
x
f (x)
1.0
2.000000
1.5
2.750000
1.8
3.440000
1.9
3.710000
1.95
3.852500
1.99
3.970100
1.995
3.985025
1.999
3.997001
x
f (x)
3.0
8.000000
2.5
5.750000
2.2
4.640000
2.1
4.310000
2.05
4.15250
2.01
4.030100
2.005
4.015025
2.001
4.003001
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) need not even 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.
3√ 2
x + 3x − 10)
a) lim (
x→1 5
b) lim (x + 3)(x2 − 6x)
x→3
c) lim
x→−2 5x2
x+3
− 2x + 7
√
d) lim t3 − t + 9
t→1
e) lim(s4 − 5s − 2)4/5
s→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.
−2x2 + x − 15
x→5
x−5
a) lim
x2 − x + 2
x→−2
x+2
b) lim
(h + 3)2 − 9
h→0
5h
c) lim
4−t
√
t→4 2 −
t
d) lim
e) lim
x→0
1
x3
6
Example 7. Let
f (x) =
x2 − 3x + 5 if x < 2
(7 − x)x
if x ≥ 2
a) Find lim− f (x) and lim+ f (x).
x→2
x→2
b) Does lim f (x) exist?
x→2
7
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.
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.
8
Any polynomial is continuous everywhere.
Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.
If f is continuous at b and lim g(x) = b, then
x→a
If g is continuous at a and f is continuous at g(a), then f (g(x)) is continuous at a.
Example 9. Explain why each function is discontinuous at the given point.
a) f (x) =
x2 − 4
at a = −2
x+2

 x2 − 4x + 3
x 6= 3
b) f (x) =
at a = 3
x
−
3
 3
x=3
9
Example 10. Let

 x+1 x<0
x2 + 1 0 ≤ x ≤ 1
g(x) =
 3
x +1 x>1
Show that g(x) is continuous on (−∞, ∞).
Example 11. Find the points at which f is discontinuous.
 √
 −x x < 0
5
0≤x≤1
f (x) =
 √
x
x>1
Example 12. Find the value of the constant c that makes h continuous on (−∞, ∞).
2
2x + c3 x < 0
h(x) =
cx2 + 5 x ≥ 0
10