limits and derivatives 2. 2.2 the limit of a function limits and derivatives in this section, we...
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LIMITS AND DERIVATIVESLIMITS AND DERIVATIVES
2
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2.2The Limit of a Function
LIMITS AND DERIVATIVES
In this section, we will learn:
About limits in general and about numerical
and graphical methods for computing them.
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Let’s 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.
THE LIMIT OF A FUNCTION
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From the table and the
graph of f (a parabola)
shown in the figure,
we see that, when x is
close to 2 (on either
side of 2), f(x) is close
to 4.
THE LIMIT OF A FUNCTION
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In fact, it appears that
we can make the
values of f(x) as close
as we like to 4 by
taking x sufficiently
close to 2.
THE LIMIT OF A FUNCTION
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We express this by saying “the limit of
the function f(x) = x2 – x + 2 as x
approaches 2 is equal to 4.” The notation for this is:
2
2lim 2 4x
x x
THE LIMIT OF A FUNCTION
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In general, we use the following
notation. 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 (on either side of a) but not equal to a.
limx a
f x L
THE LIMIT OF A FUNCTION Definition 1
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Roughly speaking, this says that the values
of f(x) tend to get closer and closer to the
number L as x gets closer and closer to the
number a (from either side of a) but x a. A more precise definition will be given in
Section 2.4.
THE LIMIT OF A FUNCTION
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An alternative notation for
is as
which is usually read “f(x) approaches L as
x approaches a.”
limx a
f x L
THE LIMIT OF A FUNCTION
( )f x L x a
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Notice the phrase “but x 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.
THE LIMIT OF A FUNCTION
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The figure shows the graphs of
three functions. Note that, in the third graph, f(a) is not defined and, in
the second graph, . However, in each case, regardless of what happens at
a, it is true that .
THE LIMIT OF A FUNCTION
( )f x L
lim ( )x a
f x L
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21
1lim
1x
x
x
THE LIMIT OF A FUNCTION Example 1
lim ( )x a
f x
Guess the value of .
Notice that the function f(x) = (x – 1)/(x2 – 1) is not defined when x = 1.
However, that doesn’t matter—because the definition of says that we consider values of x that are close to a but not equal to a.
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The tables give values
of f(x) (correct to six
decimal places) for
values of x that
approach 1 (but are not
equal to 1). On the basis of the values,
we make the guess that
THE LIMIT OF A FUNCTION Example 1
21
1lim 0.5
1x
xx
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Example 1 is illustrated by the graph
of f in the figure.
THE LIMIT OF A FUNCTION Example 1
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Now, let’s change f slightly by
giving it the value 2 when x = 1 and calling
the resulting function g:
2
11
12 1
xif x
g x xif x
THE LIMIT OF A FUNCTION Example 1
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This new function g still has the
same limit as x approaches 1.
THE LIMIT OF A FUNCTION Example 1
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Estimate the value of .
The table lists values of the function for several values of t near 0.
As t approaches 0, the values of the function seem to approach 0.16666666…
So, we guess that:
2
20
9 3limt
t
t
THE LIMIT OF A FUNCTION Example 2
2
20
9 3 1lim
6t
t
t
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What would have happened if we
had taken even smaller values of t? The table shows the results from one calculator. You can see that something strange seems to be
happening. If you try these
calculations on your own calculator, you might get different values but, eventually, you will get the value 0 if you make t sufficiently small.
THE LIMIT OF A FUNCTION Example 2
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Does this mean that the answer is
really 0 instead of 1/6? No, the value of the limit is 1/6, as we will
show in the next section.
THE LIMIT OF A FUNCTION Example 2
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The problem is that the calculator
gave false values because is
very close to 3 when t is small. In fact, when t is sufficiently small, a calculator’s
value for is 3.000… to as many digits as the calculator is capable of carrying.
THE LIMIT OF A FUNCTION Example 2
2 9t
2 9t
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Something very similar happens when we try to graph the function
of the example on a graphing calculator or computer.
2
2
9 3tf t
t
THE LIMIT OF A FUNCTION Example 2
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These figures show quite accurate graphs
of f and, when we use the trace mode (if
available), we can estimate easily that the
limit is about 1/6.
THE LIMIT OF A FUNCTION Example 2
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However, if we zoom in too much, then
we get inaccurate graphs—again because
of problems with subtraction.
THE LIMIT OF A FUNCTION Example 2
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Guess the value of .
The function f(x) = (sin x)/x is not defined when x = 0. Using a calculator (and remembering that, if ,
sin x means the sine of the angle whose radian measure is x), we construct a table of values correct to eight decimal places.
0
sinlimx
x
x
THE LIMIT OF A FUNCTION Example 3
x °
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From the table and the graph, we guess that
This guess is, in fact, correct—as will be proved later, using a geometric argument.
0
sinlim 1x
x
x
THE LIMIT OF A FUNCTION Example 3
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Investigate .
Again, the function of f(x) = sin ( /x) is undefined at 0.
0limsinx x
THE LIMIT OF A FUNCTION Example 4
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Evaluating the function for some small values of x, we get:
Similarly, f(0.001) = f(0.0001) = 0.
THE LIMIT OF A FUNCTION Example 4
1 sin 0f 1sin 2 0
2f
1sin 3 0
3f
1sin 4 0
4f
0.1 sin10 0f 0.01 sin100 0f
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On the basis of this information,
we might be tempted to guess
that .
This time, however, our guess is wrong. Although f(1/n) = sin n = 0 for any integer n, it is
also true that f(x) = 1 for infinitely many values of x that approach 0.
0limsin 0x x
THE LIMIT OF A FUNCTION Example 4
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The graph of f is given in the figure. The dashed lines near the y-axis indicate that the
values of sin( /x) oscillate between 1 and –1 infinitely as x approaches 0.
THE LIMIT OF A FUNCTION Example 4
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Since the values of f(x) do not approach a fixed number as approaches 0, does not exist.
THE LIMIT OF A FUNCTION Example 4
0limsinx x
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Find .
As before, we construct a table of values. From the table, it appears that:
3
0
cos5lim 0
10,000x
xx
3
0
cos5lim
10,000x
xx
THE LIMIT OF A FUNCTION Example 5
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If, however, we persevere with smaller values of x, this table suggests that:
3
0
cos5 1lim 0.000100
10,000 10,000x
xx
THE LIMIT OF A FUNCTION Example 5
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Later, we will see that:
Then, it follows that the limit is 0.0001.
THE LIMIT OF A FUNCTION Example 5
0lim cos5 1x x
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Examples 4 and 5 illustrate some of the
pitfalls in guessing the value of a limit. It is easy to guess the wrong value if we use
inappropriate values of x, but it is difficult to know when to stop calculating values.
As the discussion after Example 2 shows, sometimes, calculators and computers give the wrong values.
In the next section, however, we will develop foolproof methods for calculating limits.
THE LIMIT OF A FUNCTION
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The Heaviside function H is defined by:
The function is named after the electrical engineer Oliver Heaviside (1850–1925).
It can be used to describe an electric current that is switched on at time t = 0.
0 1
1 0
if tH t
if t
THE LIMIT OF A FUNCTION Example 6
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The graph of the function is shown in
the figure. 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. So, does not exist.
THE LIMIT OF A FUNCTION Example 6
0limt H t
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We noticed in Example 6 that H(t)
approaches 0 as t approaches 0 from the
left and H(t) approaches 1 as t approaches
0 from the right. We indicate this situation symbolically by writing
and . The symbol ‘ ’ indicates that we consider only
values of t that are less than 0. Similarly, ‘ ’ indicates that we consider only values
of t that are greater than 0.
0lim 0t
H t
0lim 1t
H t
ONE-SIDED LIMITS
0t
0t
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We write
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.
limx a
f x L
ONE-SIDED LIMITS Definition 2
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Notice that Definition 2 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 .
Thus, the symbol ‘ ’ means that we consider only .
limx a
f x L
ONE-SIDED LIMITS
x ax a
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ONE-SIDED LIMITS
The definitions are illustrated in the
figures.
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By comparing Definition 1 with the definition
of one-sided limits, we see that the following
is true:
lim lim limx a x a x a
f x L if and only if f x L and f x L
ONE-SIDED LIMITS
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The graph of a function g is displayed. Use it
to state the values (if they exist) of:
2
limx
g x
2
limx
g x
2
limx
g x
5
limx
g x
5
limx
g x
5
limx
g x
ONE-SIDED LIMITS Example 7
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From the graph, we see that the values of
g(x) approach 3 as x approaches 2 from the
left, but they approach 1 as x approaches 2
from the right. Therefore, and .
2
lim 3x
g x
2lim 1x
g x
ONE-SIDED LIMITS Example 7
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As the left and right limits are different,
we conclude that does not
exist.
ONE-SIDED LIMITS Example 7
2
limx
g x
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5
lim 2x
g x
5lim 2x
g x
ONE-SIDED LIMITS Example 7
The graph also shows that
and .
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For , the left and right limits are the
same. So, we have . Despite this, notice that .
5
lim 2x
g x
5 2g
ONE-SIDED LIMITS Example 7
5
limx
g x
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Find if it exists.
As x becomes close to 0, x2 also becomes close to 0, and 1/x2 becomes very large.
20
1limx x
INFINITE LIMITS Example 8
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In fact, it appears from the graph of the function f(x) = 1/x2 that the values of f(x) can be made arbitrarily large by taking x close enough to 0.
Thus, the values of f(x) do not approach a number. So, does not exist.
INFINITE LIMITS Example 8
0 2
1limx x
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To indicate the kind of behavior exhibited
in the example, we use the following
notation:
This does not mean that we are regarding ∞ as a number. Nor does it mean that the limit exists. It simply expresses the particular way in which the limit
does not exist. 1/x2 can be made as large as we like by taking x close
enough to 0.
0 2
1limx x
INFINITE LIMITS Example 8
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In general, we write symbolically
to indicate that the values of f(x) become
larger and larger—or ‘increase without
bound’—as x becomes closer and closer
to a.
limx a
f x
INFINITE LIMITS Example 8
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Let f be a function defined on both sides
of a, except possibly at a itself. Then,
means 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.
limx a
f x
INFINITE LIMITS Definition 4
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Another notation for is:
Again, the symbol is not a number. However, the expression is often read as
‘the limit of f(x), as x approaches a, is infinity;’ or ‘f(x) becomes infinite as x approaches a;’ or ‘f(x) increases without bound as x approaches a.’
limx a
f x
INFINITE LIMITS
f x as x a
lim
x af x
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This definition is illustrated
graphically.
INFINITE LIMITS
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A similar type of limit—for functions that
become large negative as x gets close to
a—is illustrated.
INFINITE LIMITS
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Let f be defined on both sides of a, except
possibly at a itself. Then,
means that the values of f(x) can be made
arbitrarily large negative by taking x
sufficiently close to a, but not equal to a.
limx a
f x
INFINITE LIMITS Definition 5
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The symbol can be read
as ‘the limit of f(x), as x approaches a,
is negative infinity’ or ‘f(x) decreases
without bound as x approaches a.’ As an example, we have:
20
1limx x
INFINITE LIMITS
limx a
f x
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Similar definitions can be given for the
one-sided limits:
Remember, ‘ ’ means that we consider only values of x that are less than a.
Similarly, ‘ ’ means that we consider only .
limx a
f x
limx a
f x
limx a
f x
limx a
f x
INFINITE LIMITS
x a
x a x a
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Those four
cases are
illustrated
here.
INFINITE LIMITS
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The line x = a is called a vertical asymptote
of the curve y = f(x) if at least one of the
following statements is true.
For instance, the y-axis is a vertical asymptote of the curve y = 1/x2 because .
limx a
f x
limx a
f x
limx a
f x
limx a
f x
limx a
f x
limx a
f x
INFINITE LIMITS Definition 6
0 2
1limx x
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In the figures, the line x = a is a vertical
asymptote in each of the four cases shown. In general, knowledge of vertical asymptotes is very
useful in sketching graphs.
INFINITE LIMITS
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Find and .
If x is close to 3 but larger than 3, then the denominator x – 3 is a small positive number and 2x is close to 6.
So, the quotient 2x/(x – 3) is a large positive number.
Thus, intuitively, we see that .
3
2lim
3x
x
x 3
2lim
3x
x
x
INFINITE LIMITS Example 9
3
2lim
3x
x
x
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Similarly, if x is close to 3 but smaller than 3, then x - 3 is a small negative number but 2x is still a positive number (close to 6).
So, 2x/(x - 3) is a numerically large negative number.
Thus, we see that .3
2lim
3x
x
x
INFINITE LIMITS Example 9
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The graph of the curve y = 2x/(x - 3) is
given in the figure. The line x – 3 is a vertical asymptote.
INFINITE LIMITS Example 9
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Find the vertical asymptotes of
f(x) = tan x. As , there are potential vertical
asymptotes where cos x = 0. In fact, since as and
as , whereas sin x is positive when x is near /2, we have:
and
This shows that the line x = /2 is a vertical asymptote.
INFINITE LIMITS Example 10
sintan
cos
xx
x
cos 0x / 2x cos 0x / 2x
/ 2lim tan
xx
/ 2lim tan
xx
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Similar reasoning shows that the
lines x = (2n + 1) /2, where n is an
integer, are all vertical asymptotes of
f(x) = tan x. The graph confirms this.
INFINITE LIMITS Example 10
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Another example of a function whose
graph has a vertical asymptote is the
natural logarithmic function of y = ln x. From the figure, we see that . So, the line x = 0 (the y-axis)
is a vertical asymptote. The same is true for
y = loga x, provided a > 1.
0lim lnx
x
INFINITE LIMITS Example 10