The concept of a limit is fundamental to Calculus. In fact, Calculus without limits is like Romeo without Juliet. It is at the heart of so many Calculus concepts like the derivative, the integral, etc. So what is a limit?

Maybe the best example to illustrate limits is through average and instantaneous speeds: Let us assume you are traveling from point A to point B while passing through point C. Then we know how to compute the average speed from A to B: it is simply the ratio between the distance from A to B and the time it takes to travel from A to B. Though we know how to compute the average speed this has no real physical meaning.

Indeed, let us suppose that a policeman is standing at point C checking for speeders going through C. Then the policeman does not care about the average speed. He only cares about the speed that you see on the speedometer, the one that the car actually has when crossing C. That one is real.

How do we compute this "instantaneous speed"? That's not easy at all! Naturally one way to do this is to compute the average speed from C to points close to C. In this case, the distance between these points and C is very small as well as the time taken to travel from them to C. Then we look at the ratio: Do these average speeds over small distances get close to a certain value? If so, that value should be called be the instantaneous speed at C. In fact, this is exactly how the policeman's radar computes the driver's speed!

Let us express this more mathematically. If *s*(*t*) is a function
that determines the position of the moving object, and assume that
at time *t*_{0}, the moving object is at *C*. At
,
we are at a point close to C. Then the average speed between
these two points is

Then we study these numbers when gets smaller and smaller. This is exactly the idea behind the concept of limit. We will write

to indicate the instantaneous speed at

Before we state the formal definition of the limit, let us
consider the function
.
What is

Clearly this function makes sense as long as the input is not equal to 0. In other words, we can take as an input any number close enough to 0, but not 0 itself.

It is clear by looking at the outputs that, when *x* gets close
to 0,
is getting close to 1. We
say that
has a limit of 1 when
*x* goes to 0 and write

You have to be very careful when you use calculators not to jump
to conclusions too quickly. Quantities may be getting close to
each other up to a certain point but then they may move further
away from each other again. This happens frequently when dealing
with chaotic systems, for example. Most of the calculators do
computations up to nine digits or so. So two numbers with the same
nine decimals are equal (according to the calculator). Be aware
of the dangers from these shortcomings of calculating devices!
But in the above statement, we mean that
is getting as close to 1 as we wish. Of
course, if you want to get close up to 75 decimals then you will
have to consider inputs *x* extremely close to 0. In other
words, for a given error
,
then if *x* is close
enough to 0, we will have
is
getting close to 1 up to
,
or equivalently

How do we express: "

**Definition of limit.** Let *f*(*x*) be a function defined
around a point *c*, maybe not at *c* itself. We have

iff for any , there exists such that

The number

Sometimes the function is not defined around the point *c* but
only to the left or right of *c*. Then we have the concepts of
left-limit and right-limits at *c*.

- (i)
*L*is the left-limit of*f*(*x*) at*c*iff for any , there exists such that

and write .- (ii)
*L*is the right-limit of*f*(*x*) at*c*iff for any , there exists such that

and write .

Of course, if a function has a limit when *x* get closer to *c*from both sides then the left and right limits exists and are
equal to the limit at the point, i.e. if
exists then

The following joke comes to my mind: *An engineer, a
physicist and a mathematician take a train ride through the
Scottish countryside. Suddenly they see a sheep outside in a
meadow. The engineer says: "Wow, in Scotland all sheep are
black!" The physicist replies: "Not really; there is at least one
black sheep in Scotland!" - The mathematician smiles and replies:
"There is at least one sheep in Scotland with at least one black
side."* (My apologies to all engineers, who seem to be at the
receiving end of most math jokes!).

What's the point? Whether you want to look at the limit world through the eyes of the "physicist" or the "mathematician" depends on your and your teacher's expectations! Maybe it suffices to stay with the "getting closer"-idea, maybe you need to dig into the workings of the formal - definition.

**Example.** Consider the function
.
We have

So obviously we have

which implies that does not exist.

**Example.** Consider the function
.
Let *x* get closer and closer to
0. For example we have

We see that *f*(*x*) does not get close to anything, even when *x*is close to 0 from the right, or the left. Thus
does not
exist. But this example is nastier than the previous one. In the
previous example, the left and right limits did at least exist,
but were not equal.

**Example.** Let
*f*(*x*) = *x*^{2}. It is easy to see that

Let us show this through the formal definition. Indeed, let . Since we want

This finishes the proof of our claim.

Note that it was quite easy in this example to find
but
in general this can be quite a challenge. One may wonder how do
we find the control number ? In general, we start our
investigation from our conclusion:
,
and
try to come up with |*x*-*c*| through some algebraic
manipulations. Let us illustrate this with an example:

**Example.** Consider the function
and *c* =
9. Then we obviously have

Indeed, let . We want to find such that if then . But

So if , then

So if we choose , or equivalently , then

In the following example, we discuss a limit at a "generic" point
*c*.

**Example.** Let *f*(*x*) =*x* and *g*(*x*) = *C*, where *C* is a
constant. Then for any point *a*, we have

and

You may want to check these two statements by going through the - definition.

**Exercise 2.** Which of the statements below are true knowing
that

- (a)
- ;
- (b)
- ;
- (c)
- If
,
then
2.99 <
*L*< 3.01.

What about the converse?

**
**

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Helmut Knaust

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