2. INTRODUCTION
Limits are the backbone of calculus and calculus
is called the mathematics of change. The study
of limits is necessary in studying change in great
detail. The evaluation of a particular limit is
what underlies the formulation of the derivative
and the integral of a function.
3. For starters, imagine that you are going to watch
a basketball game. When you choose seats, you
would want to be as close to the action as
possible. You would want to be as close to the
players as possible and have the best view of the
game, as if you were in the basketball court
yourself. Take note that you cannot actually be
in the court and join the players, but you will be
close enough to describe clearly what is
happening in the game.
4. This is how it is with limits of functions . We will
consider functions of a single variable and study
the behavior of the functions as its variable
approaches a particular value ( a constant). The
variable can only take values very, very close to
the constant but it cannot equal the constant
itself. However, the limit will be able to describe
clearly what is happening to the function near the
constant.
5. Limit
Consider a function f of a single variable x.
Consider a constant c which the variable x will
approach (c may or may not be in the domain of f).
The limit, to be denoted by L, is the unique real
value that f(x) will approach as x approaches c. In
symbols, we write the process as
lim f(x) = L
x c
This is read, “ The limit of f(x) as x approaches c is
L.”
6. Consider lim ( 1 + 3x)
x 2
First, consider approaching 2 from its left
through values less than 2.
x f(x)
1 4
1.4 5.2
1.7 6.1
1.9 6.7
1.95 6.85
1.997 6.991
1.9999 6.9997
1.9999999 6.9999997
7. Consider approaching 2 from its right or through
values greater than but close to 2.
x f(x)
3 10
2.5 8.5
2.2 7.6
2.1 7.3
2.03 7.09
2.009 7.027
2.0005 7.0015
2.0000001 7.0000003
8. As the values of x get closer and closer to 2, the
values of f(x) get closer and closer to 7. This
behavior can be shown no matter what set of
values, or what direction taken in approaching 2.
In symbols,
lim ( 1 + 3x ) = 7.
x 2
9. Example 1: Investigate
limit (x2 + 1)
x - 1
By constructing table of values. Here, c= - 1 and
f(x) = x2 + 1.
We start again by approaching – 1 from the left.
x f(x)
- 1. 5 3. 25
- 1. 2 2. 44
- 1. 01 2. 0201
- 1. 0001 2.00020001
10. Now approach – 1 from the right.
The tables show that as x approaches – 1, f(x)
approaches 2. In symbols,
lim (x2 + 1 ) = 2 .
x - 1
x f(x)
- 0. 5 1.25
-0. 8 1. 64
- 0. 99 1.9801
- 0. 9999 1. 99980001
11. Example 2: Investigate lim /x/ through
x 0
a table of values.
Approaching 0 from the left and from the right,
we get the following tables:
Hence, lim /x/ = 0
x 0
x /x/
- 0.3 0.3
-0.01 0.01
-0.00009 0.00009
-0.00000001 0.00000001
x /x/
0.3 0.3
0.01 0.01
0.00009 0.00009
0.00000001 0.00000001
12. Example 3. Investigate
lim x2 – 5x + 4
x 1 x – 1
Approach 1 from the left
x f(x)
1.5 -2.5
1.17 -2.83
1.003 -2.9977
1.00-1 -2.9999
13. Approach 1 from the right
The tables show that as x approaches 1, f(x)
approaches – 3. In symbols,
lim x2 – 5x + 4 = - 3
x 1 x – 1
x f(x)
0.5 -3.5
0.88 -3,12
0.996 -3.004
0.9999 -3.0001
