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Limits and Derivatives
Limit of a Function
Let’s investigate the behavior of the function f defined by f (x) = x^2 − 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.
investigate the behavior of the function defined by
f near 2. The following table gives values of for values of clo
qual to 2.
x f x x
f f x x^2 x
SECTION 2.2 THE LIMIT OF A FUNCTION
x
x^ f^ x
f x
From the table and the graph of f (a parabola) shown in Figure we see that when x is close to 2 (on either side of 2), f (x) is close to 4. We express this by saying “the limit of the function f (x) = x^2 − x + 2 as x approaches 2 is equal to 4.” The notation for this is
lim x→ 2
(x^2 − x + 2) = 4.
4
ƒ approaches
x
y
2 As x approaches 2,
y=≈- x+
0
FIGURE 1
An alternative notation for
x^ lim→a f^ (x) =^ L
is f (x) → L as x → a
which is usually read “f (x) approaches L as x approaches a.”
Notice the phrase “but x 6 = a” in the definition of limit.
This means that in finding the limit of f (x) as x approaches a, we never consider x = a.
In fact, f (x) need not even be defined when x = a. The only thing that matters is how f is defined near a.
(a)
x
y
L
a
Figure 1:
(c)
x
y
L
a
Figure 3:
Figures show the graphs of three functions. Note that in Figure 3, f (a) is not defined and in Figure 2, f (a) 6 = L. But in each case, regardless of what happens at a, (^) xlim→a f (x) = L
Solution (cont.)
(- 1,0.841471) (1,0.841471)
1
x
y
From the table and the graph in Figure we guess that
lim x→ 0
sin x x
This guess is in fact correct, we will prove later using a geometric argument.
Solution (cont.)
From the table and the graph in Figure we guess that
lim x→ 0
sin x x
This guess is in fact correct, we will prove later using a geometric argument.
Solution (cont.)
On the basis of this information we might be tempted to guess that
lim x→ 0
sin
π x
but this time our guess is wrong. Note that although f (1/n) = sin nπ = 0 for any integer n, it is also true that f (x) = 1 for infinitely many values of x that approach 0. [In fact, sin(π/x) = 1 when
π x
π 2
and, solving for x, we get x = 2/(4n + 1).]
Solution (cont.)
The graph of f is given in Figure.
The broken lines indicate that the values of oscillate between 1 and infinitely often as approaches 0. (Use a graphing device to graph and zoom in toward the origin several times. What do you observe?) Since the values of f x do not approach a fixed number as x approaches 0,
x f
sin x 1
y=sin(π/x)
x
y 1
1
_
_
SECTION 2.2 THE LIMIT OF A FUNCTION 105
The broken lines indicate that the values of sin(π/x) oscillate between 1 and − 1 infinitely often as x approaches 0.
Example 4
The Heavisidea^ function H is defined by,
H(t) =
0 , t < 0 1 , t ≥ 0
Its graph is shown in Figure.
EXAMPLE 6 The Heav
[This function is nam and can be used to de graph is shown in Fig As approaches 0 right, approache approaches 0. Theref
H t
t
106 CHAPTER 2 LIMITS AND DERIVATIVES
FIGURE 8
t
y
1
0
aThis function is named after the electrical engineer Oliver Heaviside
(1850–1925) and can be used to describe an electric current that is switched on at time t = 0.
MAT 1001 Calculus I 15 / 77
EXAMPLE 6 T
[This functio
and can be u
graph is sho
As appr
right, a
approaches 0
H t
t
106 CHAPTER 2 LIMITS AND DERIVATIVES
FIGURE 8
t
y
As t approaches 0 from the left, H(t) approaches 0. As t approaches 0 from the right, H(t) approaches 1. There is no single number that H(t) approaches as t approaches 0. Therefore, lim t→ 0
H(t) does not exist.
Limit of a Function One-Sided Limits
Definition 5
We write lim x→a−^
f (x) = 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] 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.
Limit of a Function One-Sided Limits
Notice that Definition 5 differs from Definition 1 only in that we require x to be less than a.
Similarly, if we require that x be greater than a, we get “the right-hand limit of f (x) as x approaches a is equal to L” and we write
lim x→a+^
f (x) = L
Thus, the symbol “x → a+” means that we consider only x > a.
These definitions are illustrated in Figure 4.
