OBJECTIVE
To find analytically
lim f(x) =
PRE-REQUISITE KNOWLEDGE
Concept of limits
Concept of left hand and right hand limits
MATERIALS REQUIRED
Pen/Pencil
Graph paper
" White sheets of paper
" Geomnetry box
Calculator, etc.
PROCEDURE
1. Consider the function f given by fx) =
*-9
(x-3)\x +3)
x-3
x-3
= t+3.
f(x) ie., lim3
2. The function is not defined at x = 3. We are to find lim
*3
-9
-3
3.
3. Take some values of x less than 3 and some other values of x more than
4. In both the cases, the values to be taken have to be very close to 3.
5. Calculate the corresponding values of f at each of the values of x taken close to 3.
6. Write the values of fr) in the following tables:
Table 1
S’3
2.9
2.99
2.999
2.9999
2.99999
2.999999
f)
5.9
5.99
5.999
5.9999
5.99999
5.999999
Table 1 shows the values off ) as * ’3and * < 3, i.e., x ’ 3.
Thus, the value of fr) approaches 6.
120
AFU
LIMITOF
Table 2
3.1
3.01
3.001
3.0001
3.00001
3.000001
6.1
6.01
6.001
6.0001
6.00001
6.000001
Table 2 shows the values of f«) as x ’3 and x> 3, i.e., z’ 3.
Thus, the value of fa) approaches 6.
7.Take a rectangular piece of graph paper.
&. Draw the graph of fu) on the graph paper as shown in Fig. 21.1.
8
5
3x3
EX'AO
X
59
Fig. 21.1 Graph of ) =
-9
X-3
OBSERVATIONS
1. Values of fr) as x - 3 from the left on the number line, as in Table 1, are coming
closer and closer to
2. Values of fx) as z ’ 3* from the right on the number line, as in Table 2, are coming
closer and closer to
From observations (1) and (2), lim f(r) =13*
lim f(z) =
RESULT
lim - a
Dy this activity, we have analytically explained the concept of limit
When x ’ a.
MANUAL-X
122
PRACTICAL IMPORTANCE
takes, how likely am Ito
it
as
long
as
coin
a
1. IfI keep tossing
say
never
toss a head?
problem, we might
as a'n'limit
Rephrased
Ifltoss a coin
times, what is the probability P(n) that I have not yet tossed a head?
oo of P(n)?
What is the limit as n
answer to this is
The mathematical
lim P(n) = lim
n’o
Then,
n’oo
11 1
1
gets closer and closer to zero as n gets "closer to oo'.
because P= 2' 4' 8' 16
Suppose that the money in vev
interest.
of
compounding
continuous
is
example
2. A good
se
of 'r and it is compounded 'n' times ner
bank account has an annual interest rate
account, then after t years your money L
your
in
rupees
P'
had
initially
you
If
grown to
periods becomes infinite, At
In continuous compounding, the number of compounding
earn in each one of those eriods
the same time, the fractional rate of interest you
becomes infinitesimal or infinitely small.
We know that the formula for compound interest is
A =P
limit:
Let's derive the formula for continuous compound interest by computing the
lim
=P lim
n’o
=P lim
Since, e is an irrational number, it is defined as
e = lim
n ’ oo
= Pert