unit 1 differential calculus - IGNOU

Solution
Look at Tables 1.1 and 1.2. These give values of f (x) as x gets closer and
closer at 2 through values less than 2 and through values greater than 2,
respectively.
Table 1.1
x 1 1.5 1.9 1.99 1.999
f (x) 5 5.5 5.90 5.99 5.999
Table 1.2
x 3 2.5 2.1 2.01 2.001
f (x) 7 6.5 6.1 6.01 6.001
From the above tables, it is clear that as x approaches 2, f (x) approaches 6.
In fact, the nearer x is chosen to 2, the closer f (x) will be to 6. Thus, 6 is
limit of x + 4 as x approaches 2, that is, ( x + 4)
= 6.
lim
x → 2
In the above example, the value of ( + 4)
x coincides with the value
x + 4 when x = 2, that is,
lim
x →2
lim
x →2
f ( x)
= f
Numbers x near 2 fall into two natural categories; those which are < 2, that
is, those that lie to the left of 2, and those which are > 2, that is, those which
lie to the right of 2.
We write
lim
−
x →2
f ( x)
= 6
( 2).
to indicate that as x approaches 2 from the left, f (x) approaches 6.
We shall describe this limit as the left-hand limit of f (x) as x approaches (or
tends to) 2.
Similarly,
lim
+
x →2
f ( x)
= 6
indicates that as x tends to 2 from the right, f (x) approaches 6.
We shall call this limit as the right-handed limit of f (x) as x approaches 2.
The left and right-hand limits are called one-sided limits.
It is clear now that
if and only if, both
lim
x →2
lim
−
x →2
f ( x)
= 6
f ( x)
= 6
lim
x →2
and
lim
+
x →2
f ( x)
= 6 .
In the above example, the value of ( + 4)
x coincides with the value of
x + 4 when x = 2, that is
lim
x →2
f ( x)
= f
( 2).
Differential Calculus
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