Calculus Of One Real Variable By Pheng Kim Ving

Chapter 1: Limits And Continuity Section 1.1.1: Limits

 

1.1.1
Limits

 

 

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 1. Origin Of The Concept Of Limits

 

Tangents

 

A tangent to a circle is a straight line that intersects the circle at a single point. See Fig. 1.1. So can we define a tangent
to an arbitrary curve as a straight line that intersects the curve at a single point? To find out let's look at Fig. 1.2. Clearly

 

Fig. 1.1

 

Tangent to a circle intersects circle at a single point.

 

Fig. 1.2

 

T1: a tangent,
T2: a tangent,
S: not a tangent.

 

line T1 intersects curve C at a single point and is a tangent to C, line T2 intersects C at more than 1 point and is tangent
to
C, and line S intersects C at a single point and isn't a tangent to C. Thus we can't define a tangent to a curve as a
straight line that intersects the curve at a single point.

 

To define a tangent to a curve we proceed as follow. Let C be a curve and P a point on it. Refer to Fig. 1.3. Let's define
the tangent to
C at P. Let S be a secant to C thru P and intersecting C at another point Q. Let Q get closer and closer
to, or approach or tend to,
P. See Fig. 1.4, where Q is on one side of P, and Fig. 1.5, where Q is on the other side of P.
Points
Q1, Q2, and Q3 and secants S1, S2, and S3 represent 3 positions of Q and the 3 corresponding positions of S
respectively as
Q approaches P. For this curve C and this point P, as Q approaches P from one side, S approaches a
 

Fig. 1.3

 

Using Secant S To Define Tangent
T.

 

Fig. 1.4

 

As Q approaches P from one side,
S approaches T.

 

Fig. 1.5

 

And as Q approaches P from
the other side,
S also
approaches
T.

 

line T, and as Q approaches P from the other side, S also approaches the same line T. The tangent to C at P is defined
to be
T. We can make S arbitrarily close to T (meaning as close to T as we like) by making Q sufficiently close to P. This
situation was a part of the origin of the concept of limits. We say that
T is the limit  of S or that S approaches T as Q
approaches
P. The tangent to C at P is defined to be the limit of secant PQ as Q approaches P. Here “approaches”
means approaches from both sides. When there's no specifying of side, it's understood that both sides must be
considered. The limit is a unique line that
PQ approaches as Q approaches P from both sides of P.

 

To see why we require that both sides must be considered, look at Fig. 1.6. Curve C changes direction smoothly at every
point of it except at point
P, where it changes direction abruptly. Now look at Fig. 1.7. As Q labelled Q1 approaches P

Fig. 1.6

 

C changes direction abruptly at P.

 

Fig. 1.7

 

L1 and L2 are different. No tangent to C
at
P.

 

from one side of P, secant PQ labelled PQ1 approaches line L1, and as Q now labelled Q2 approaches P from the other
side of
P, secant PQ now labelled PQ2 approaches line L2, which is different from L1. Now if a tangent exists then it must
be unique. Consequently
L1 and L2 can't be tangents to C at P. If we consider only 1 side of P then we'll be led to the
wrong conclusion that either
L1 or L2 is a tangent to C at P. That's why we require that both sides must be considered.

 

Slopes

 

Refer to Fig. 1.3. As Q approaches P, secant PQ approaches tangent T, so naturally the slope of PQ approaches the
slope of
T. The slope of the tangent at P is defined to be the limit of the slope of secant PQ as Q approaches P.

 

Rates Of Change

 

Imagine there's a road having a portion that's a straight line measuring a little more than 300 km. A car travelling on it
passes point
A at 1:00 pm, point B, which is east of A, at 3:00 pm on the same day, and point C, which is east of B, at
5:00 pm on the same day. See Fig. 1.8. The distances
AB and AC are 100 km and 300 km respectively. The portion
AC is a straight line.

 

Take the direction from west to east of the road as positive. Choose A as the initial position s = 0 of the car. So B is at
position
s = 100 and C at position s = 300, where s is in km. Since AC is a straight line, any position to the right of A up
to
C can be determined by a single number, the positive distance from A to that position (positive because of being east
of
A). Select 1:00 pm when the car passes A as the initial time t = 0. Thus 3:00 pm corresponds to t = 2 and 5:00 pm to
t = 4, where t is in h (hours). Position s is a function of time t: s = s(t). We have s(0) = 0, s(2) = 100, and s(4) = 300.

 

If the car travels from B to C at a constant velocity, then its average velocity over time interval [2, 4] is:

 

and its velocity at any instant or point in that time interval, which exists as evidenced by the speedometer, is also
100 km/h. Now suppose it travels from
B to C at a velocity that sometimes changes. Its average velocity over [2, 4] is its

Fig. 1.8

 

Velocity And Rate Of change.


velocity that it would have if it travelled at a constant velocity over that time interval. As a consequence that average
velocity is still (
s(4) – s(2))/(4 – 2) = (300 – 100)/2 = 100 km/h. However its velocity at any particular instant in that time
interval may be different from its average velocity over that time interval.

 

The car changes position from B to C by 200 km in 2 h. Hence in average it changes position by 200/2 = 100 km per h,
written as 100 km/h. As the hour is a unit of time, this is the average change of position per 1-unit change of time. It's
called the average rate of change of position with respect to time. The minute, second, or day are also units of time.
Position is a function of time. In general the average rate of change of a function
y = f(x) over [x1, x2] (with respect to x)
is the average change of the function
f per 1-unit change of the variable x in [x1, x2], given by the ratio ( f(x2) – f(x1))/
(
x2x1). At any point or instant x = a in dom( f ), f also has a rate of change, called instantaneous rate of change of f at
a, or simply rate of change  of f at a, because a is a point or instant and therefore the rate of change of f at it can't be
anything other than instantaneous.

 

Consider the function y = f(x) = x2. See Figs. 1.9 and 1.10. In dom( f ) let a be a given point and x an arbitrary point
different from
a. It may be that x > a as in Fig. 1.9 or x < a as in Fig. 1.10. The rate of change of f (with respect to x)
is its change corresponding to a 1-unit change of
x. The average rate of change rax of f over the interval [a, x] is defined
as follows:

 

 

We also want to define the rate of change ra of f at just point a, not over an interval. It's called the instantaneous rate of
change
 of f at a, to emphasize that it's the rate at an instant or a point, not over an interval. It' also simply called the

Fig. 1.9

 

Instantaneous rate of change of f at a is
equal to limit, which is a unique number,
of average rate of change of
f over [a, x]
as
x approaches a from the right ...
(Continued to Fig. 1.10)

 

Fig. 1.10

 

(Continued from Fig. 1.9) ... and from the
left.

 

rate of change of f at a, for the reason seen above. Now how can we define ra? If there's an interval, say [a1, a2], that
contains
a and such that at each point of it the rate of change is constant, then ra is equal to the average rate of change
over [
a1, a2]. Otherwise ra generally isn't equal to the average rate of change over any interval that contains a. In this
case, should
ra be defined as ( f(a) _ f(a))/(a _ a), imitating the definition of the average rate of change over an interval?
No, because we can't divide anything by 0, not even 0 by 0. However we can choose an interval, say [
a3, a4], that
contains
a and that is small enough so that the rate of change at each point of it is approximately the same. Then ra is
approximately equal to the average rate of change over [
a3, a4]. Better still, we can proceed as follows.

 

Let x get closer and closer to a, from the right of a then from the left, but remain different from it, and see what
happens. As
x gets closer and closer to a, the interval [a, x] becomes smaller and smaller toward almost being point a,
rax gets closer and closer to ra , and the value a + x of rax gets closer and closer to a + a = 2a. So, it's reasonable to
accept that the value of
ra is 2a. Thus, let's define ra to be the quantity that rax gets closer to, or approaches or tends to,
as
x gets closer to, or approaches or tends to, a, and we obtain ra = 2a, as desired. Note that in this case the
instantaneous rate of change is exactly, not just approximately, equal to 2
a. This situation was another part of the origin
of the concept of limits. The limit  of
rax as x approaches a is defined to be the quantity that rax approaches as x
approaches
a. This yields:

 

ra = ( limit of rax as x approaches a) = 2a.

 

Note that we require that the limits from both sides of a exist and are equal in order for the limit at a to exist, as it's
defined to be their common value, and the rate of change of
f at a is defined to be it. Similarly to the case of the tangent
which must be unique if it exists, the rate of change must also be unique if it exists. For example a car on a road can't
have more than 1 different velocity relative to that road at the same time.

 

Slopes And Rates Of Change

 

Let f be a function. See Fig. 1.11. Let a and u be 2 distinct points in dom( f ), P = (a, f(a)), Q = (u, f(u)), and T the
tangent to the graph of
f at P. We have:

 

 

So slope of secant equals average rate of change and slope of tangent equals instantaneous rate of change.

 

Fig. 1.11

 

Slope Of Secant = Average Rate Of Change,
Slope Of Tangent = Instantaneous Rate Of Change.

 

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 2. Limits

 

The limit of 1/(x – 2) as x approaches 0 is –1/2. Now consider the limit of 1/(x – 2) as x approaches 2. As x approaches
2 from the left, which means
x approaches 2 and is < 2, x – 2 approaches 0 and is < 0, so 1/(x – 2) is < 0 and gets
larger and larger without bound. As
x approaches 2 from the right, which means x approaches 2 and is > 2, x – 2
approaches 0 and is > 0, so 1/(
x – 2) is > 0 and gets larger and larger without bound. Thus, the limit of 1/(x – 2) as x
approaches 2 doesn't exist. In general, the limit of
f(x) as x approaches a may or may not exist.

 

For the tangent to the graph of f(x) and the instantaneous rate of change of f at x = a to exist and be unique, both the
limits of (
f(x) – f(a))/(xa) as x approaches a from the right and from the left must exist and be equal, so that the
2-sided limit as
x approaches a from both sides exists and is unique, as it's defined to be their common value, and the
slope of the tangent and the instantaneous rate of change will be equal to this 2-sided limit. When we talk of a limit
without specifying from the left or from the right, we refer to the 2-sided limit that must be obtained equally from both
sides. It's this type of limit that we discuss in this section.

 

Now let's define limits formally.

 

 

Fig. 2.1

 

Limit of f(x) as x approaches a is L.

 

approaches 5 is 6. We remark that we can make h(x) as close to 6 as we like by making x sufficiently close to 5. In
general we say that the limit of
f(x) as x approaches a is L if we can make the value f(x) as close to L as we like by
making
x sufficiently close to a. This means that we can make the distance | f(x) _ L| from f(x) to L as small as we like
by making the distance
|x _ a| from x to a sufficiently small.

 

Definition 2.1

        

 

 

 

In the last line in the above definition, it's clear that x must also be in dom( f ) because f(x) must exist. The line is a
simplified form of this statement:

 

 

Remarks 2.1

 

 

Fig. 2.2

 

 

 

Fig. 2.3

 


L = f(a);
graph of
f   touches” point (a, L) and contains it.

 

Fig. 2.4

 


graph of f   touches” point (a, L) but doesn't contain it.

 

Fig. 2.5

 


f(a) doesn't exist;
graph of f   touches” point (a, L) but doesn't contain it.

 

     graph of f near the point (a, L) that satisfies these 2 properties: (1) it extends from the point (a, L) to both sides of
     that point (or one side if the limit is one-sided, as we'll see in a later section), and (2) it's solid (ie, it has no holes or
     jumps inside it) except it may possibly have a hole at the point (a, L). The graph of f may or may not have holes or
     jumps in it, small or large. But it must have a piece of it that satisfies those 2 properties. It doesn't matter how small
     the piece is, as long as it satisfies the 2 properties. We can intuitively think of the graph of f like this: it “ touches” the
     point (a, L) but may or may not contain it.

 

 

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 3. Proving A Limit Using Its Definition

 

Example 3.1

 

 

Solution

EOS

 

Example 3.2

 

Prove that:

 

 

Solution

EOS

 

Remarks 3.1

 

i.  When you're asked to prove a limit using the definition of the limit, you must just do that: use the definition of the
  
 limit, even if the limit is intuitively “ obvious”.

 

 

vi.  The definition of the limit provides a means of proving whether or not a particular number is the limit of  a
      particular function at a particular point. It provides no means of finding an unknown limit.

 

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 4. Why Bother With The Definition Of The Limit?

 

 

     when x gets closer and closer to a, f(x) gets closer and closer to L.” ?”,

  
 you propose.

 

OK. Let's see. Suppose we take the phrase gets closer and closer to  to mean approaches  or tends to. In the definition,
we define what approaches
 or tends to  means. If we replace the last line of the definition with the “ proposed” line, then
the definition will be incomplete, because
 gets closer and closer to  simply means approaches  or tends to, and thus
won't have been defined yet.

 

Now suppose we don't take gets closer and closer to  to explicitly mean approaches  or tends to, but we simply employ it
in its ordinary everyday meaning without defining what it means mathematically. Well, in this case, trouble may be
waiting for us. For example, let:



See Fig. 4.1. When
x gets closer and closer to 0 from the right, f(x) gets closer and closer to 0.001 from above; since
0.001 is close to 0, we can say that
f(x) “ gets closer and closer to 0” from above (from above because 0.001 > 0) (if this
 

Fig. 4.1

 

The limit of f(x) as x approaches 0 isn't 0. After studying a later section
we'll know that such a limit doesn't exist.

 

fails to convince you, replace 0.001 with 0.000,001). When x gets closer and closer to 0 from the left, f(x) gets closer
and closer to
_ 0.001 from below; since _ 0.001 is close to 0, we can say that f(x) “ gets closer and closer to 0” from
below (from below because
_ 0.001 < 0) (if this fails to convince you, replace _ 0.001 with _ 0.000,000,001).
Consequently, we can say that when
x gets closer and closer to 0, f(x) “ gets closer and closer to 0”. This leads us to
conclude that
f(x) “approaches 0” as x approaches 0. Well, clearly that conclusion is wrong, because f(x) never gets
into the interval (
_ 0.001, 0.001), which contains 0, and it follows that f(x) can't approach 0. The word limit  is the name
given to the number that the function approaches (or tends to or gets closer and closer to).

 

We can use Definition 2.1 to prove formally that:

 

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 Problems & Solutions

 

 

Solution

 

 

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Solution

 

 

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3.  Prove that:

   

     utilizing the definition of the limit.

 

Solution

 

 

That completes the proof.

 

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4.  Prove that:

 

   

 

     employing the definition of the limit.

 

Solution

 

 

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Solution

 

 

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