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Unit 7: Limits

Often, the method of direct substitution cannot be used because a is not in the domain of f. In Notes
these cases, it is sometimes possible to factor the function and eliminate terms so that the function

is defined at the point a.
Consider the function f(x)=

You can see that the function f(x) is not defined at x = 1 as x–1 is in the denominator. Take
the value of x very nearly equal to but not equal to 1 as given in the tables below. In this case
x–1 0 as x 1.

We can write f(x) = = = x + 1, because x–1 0 and so division by

(x – 1) is possible.
x f(x) x f(x)
0.5 1.5 1.9 2.9
0.6 1.6 1.8 2.8
0.7 1.7 1.7 2.7
0.8 1.8 1.6 2.6
0.9 1.9 1.5 2.5
0.91 1.91 : :
: : : :
: : 1.1 2.1
0.99 1.99 1.01 2.01
: : 1.001 2.001
: : : :
0.9999 1.9999 : :
1.00001 2.00001
In the above tables, you can see that as x gets closer to 1, the corresponding value of f (x) also gets
closer to 2.
However, in this case f(x) is not defined at x = 1. The idea can be expressed by saying that the

limiting value of f(x) is 2 when x approaches to 1.
Let us consider another function f (x) = 2x. Here, we are interested to see its behavior near the
point 1 and at x = 1. We find that as x gets nearer to 1, the corresponding value of f (x) gets closer

to 2 at x = 1 and the value of f (x) is also 2.

So from the above findings, what more can we say about the behaviour of the function near
x = 2 and at x = 2 ?
In this unit we propose to study the behaviour of a function near and at a particular point where

the function may or may not be defined.
7.2 Limits of a Function

In the introduction, we considered the function f(x) = We have seen that as x approaches l,
f (x) approaches 2. In general, if a function f (x) approaches L when x approaches ‘a’, we say that
L is the limiting value of f (x).
Symbolically it is written as
= L

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