8_Limits

DigiPen (USA) Corp. c 2013

Introduction to Limits

This lecture will serve as an introduction to the concept of limits in mathematics. A general

definition of a function’s limit is as follows:

For a function f(x), we say that the limit of f(x) as x approaches “c” is L, if we can get

values of f(x) that are as close as possible to L by choosing the values of x that are the

closest to “c”, but not equal to “c”. In mathematical notation this is expressed as

lim�→� �� = �

So what the above formula means is that as x reaches the value of “c”, f(x) will get to the value

of L.

Now that we have established the definition of a function’s limit as its variable approaches as

certain constant value, let us present some properties of limits. Let us first assume that

lim�→� �� & lim�→� � both exist:

1- lim�→�� ± � = lim�→� �� ± lim�→� �

2- lim�→� �� . � = lim�→� �� . lim�→� �

3- lim�→��

��= ��→� ��

��→� �� with the condition that lim�→� � ≠ 0

4- lim�→��

= ∞

5- lim�→��

= 0

6- If �� is an algebraic function and �� is defined, then lim�→� �� = ��. In such a

case, the function f is continuous. (Recall: an algebraic function is a function that satisfies

a polynomial equation which has terms that are themselves polynomials with rational

coefficients).

A few examples will help clarifying the idea of limits.


Example 1:

lim�→�

√ + " + 4 − 1 − 3

=√0 + 0 + 4 − 1

0 − 3= −

13

This was a very straightforward example. The next examples will deal with slightly more

complicated problems that will be solved using some tricks.

Example 2:

lim�→'�

� − 1 + 1

= lim�→'�

� − 1� + 1 + 1

= lim�→'�

� − 1 = −2

Notice that if we were to substitute -1 by x, we would get �� which is undefined. So one trick is to

look at the numerator. It is clear that the numerator is of the form − ) which can be factored

into � − )� + ). This trick helped us get rid of the denominator because it is also present in

the numerator. Hence what we are left with is to find the limit of the function − 1 as → −1,

which is easy since all we have to do is replace x by -1, because f(-1) for �� = − 1 is

defined.

Example 3:

lim�→�

* −

= lim�→�

�+ − 1 = lim

�→� �+ − 1 = −1

Again, if we were to replace x = 0 in the above left-most equation, we would get 0/0 which is

undefined. So a quick look at the numerator shows that is a common factor. We take it out

and it cancels out with the denominator. This leaves us to find the lim�→�

�+ − 1 = ��0 since f(0)

is defined in that case.

Example 4:

lim�→'

− 2 − 8 − 4

= lim�→'

� + 2� − 4� + 2� − 2

= lim�→'

− 4 − 2

=−6−4

=32


In this example, we had to factorize both the numerator and the denominator to be able to find

some common roots that can be cancelled out before finding the limit.

Example 5:

lim�→�

√ + + 23 − 5 − 1

= lim�→�

√ + + 23 − 5 − 1

∗√ + + 23 + 5

√ + + 23 + 5

= lim�→�

+ + 23 − 25

� − 10√ + + 23 + 51

= lim�→�

+ − 2

� − 10√ + + 23 + 51

= lim�→�

� − 1� + 2

� − 10√ + + 23 + 51

= lim�→�

� + 2

0√ + + 23 + 51

=1 + 2

√1 + 1 + 23 + 5=

310

= 0.3


Tangents

Given a function �� and a point 2�, ��. Given another point situated at + ℎ where h is a

very small value. The mapping of + ℎ would be �� + ℎ. Let us call this point 5� +ℎ, �� + ℎ.

Let us now connect the two points A & B through a straight line, whose slope is

6 =∆8∆

=0�� + ℎ − ��1

+ ℎ − =

0�� + ℎ − ��1ℎ

All this is shown in Figure 1 below.

Figure 1: Graphical representation of how to get the tangent

Now, what would happen when h gets closer to zero, is that + ℎ gets closer to . So point B

would be moving in the direction of point A. The line AB is what is called the secant line, which

is nothing more than a line that joins two points that intersect the same curve (f(x) in our case).

Why do we need this? Because this is how we can approximate the tangent to a curve at a point

A.

Before moving along, it is important to define what a tangent is. If you have a curve and a point

P on that curve, the tangent to that curve at point P is the straight line which just touches the

curve at that point, as is shown in Figure 2.


Going back to the point where we were moving point B towards point A by letting h approach

zero. The tangent would occur when B approaches A and the secant line reaches a limiting point.

The slope of this tangent line would be mathematically written as

6 = lim9→�

0�� + ℎ − ��1ℎ

Incidentally, this slope is also the derivative of f(x) at x, but this is a topic that will be covered in

the next course.

Figure 2: Tangent and secant lines

Example:

We need to find the equation of the line that is tangent to the curve �� = at point (2,4). To

do so, we apply the formula above with = 2.

6 = lim9→�

0�� + ℎ − ��1

ℎ= lim

9→�

� + ℎ −

ℎ

= lim9→�

��2 + ℎ − 2 ℎ


= lim9→�

4 + 4ℎ + ℎ − 4ℎ

= lim9→�

4ℎ + ℎ

ℎ= lim

9→��4 + ℎ = 4

So the equation of the tangent line at point (2,4) is:

8 − 4 = 4� − 2 → 8 = 4 − 4

8_Limits

Documents

Transcript of 8_Limits