Calculus Of One Real
Variable – By Pheng Kim Ving Chapter 1: Limits And Continuity – Section 1.1.1: Limits 
1.1.1 
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 T_{1}: a tangent, 
line T_{1} intersects curve C at a single point
and is a tangent to C, line T_{2} 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 Q_{1}, Q_{2}, and Q_{3} and secants S_{1}, S_{2}, and S_{3} 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 

Fig. 1.4 As Q approaches P from one side, 

Fig. 1.5 And as Q approaches P from 
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 Q_{1} approaches P

Fig. 1.6 C changes
direction abruptly at P. 

Fig. 1.7 L_{1} and L_{2} are different.
No tangent to C 
from one side of P, secant PQ labelled PQ_{1} approaches line L_{1}, and as Q now labelled Q_{2} approaches P from the other
side of P, secant PQ now labelled PQ_{2} approaches line L_{2}, which is
different from L_{1}. Now if a tangent
exists then it must
be unique. Consequently L_{1} and L_{2} 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 L_{1} or L_{2} 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 1unit 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 [x_{1}, x_{2}] (with respect to
x)
is the average change of the function f per 1unit change
of the variable x in [x_{1}, x_{2}], given by the
ratio ( f(x_{2}) – f(x_{1}))/
(x_{2} – x_{1}). 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) = x^{2}. 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 1unit change of x. The average
rate of change r_{ax} of f over the interval [a, x] is defined
as follows:
We also want to define the rate of change r_{a} 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 

Fig. 1.10 (Continued from Fig. 1.9) ... and from the 
rate of change of f at a, for the reason
seen above. Now how can we define r_{a}? If there's an
interval, say [a_{1}, a_{2}], that
contains a and such that at each point of it the rate of change is constant, then r_{a} is equal to the average rate of change
over [a_{1}, a_{2}]. Otherwise r_{a} generally isn't
equal to the average rate of change over any interval that contains a. In this
case, should r_{a} 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 [a_{3}, a_{4}], that
contains a and that is small enough so that the rate of change at each point of it
is approximately the same. Then r_{a} is
approximately equal to the average rate of change over [a_{3}, a_{4}]. 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,
r_{ax} gets closer and closer to r_{a} , and the value a + x of r_{ax} gets closer and closer to a + a = 2a. So, it's reasonable to
accept that the value of r_{a} is 2a. Thus, let's
define r_{a} to be the quantity that r_{ax} gets closer to,
or approaches or tends to,
as x gets closer to,
or approaches or tends to, a, and we obtain r_{a} = 2a, as desired. Note that in this case the
instantaneous rate of change is exactly, not just approximately, equal
to 2a. This situation was another part of the origin
of the concept of limits. The limit of r_{ax} as x approaches a is defined to be the quantity that r_{ax} approaches as x
approaches a. This yields:
r_{a} = ( limit of r_{ax} 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, 
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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))/(x – a) as x approaches a from the right
and from the left must exist and be equal, so that the
2sided 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
2sided limit. When we talk of a limit
without specifying from the left or from the right, we refer to the 2sided
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:

Fig. 2.2 
Fig.
2.3

Fig.
2.4

Fig.
2.5

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 onesided, 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 
Prove that:
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 
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:
Problems & Solutions 
Solution
Solution
3.
Prove that:
utilizing the definition of the
limit.
Solution
That completes the proof.
4. Prove that:
employing the definition of the limit.
Solution
Solution
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