Understanding the Concept of Limits

Maths Tutor

10 years ago

We know division by zero is not possible in mathematics. If we consider the function definition as

The value of f(x) at x=1 is indeterminate.

More simply, the value of the function f(x) does not exist at x=1. So, instead of x=1 we consider values of x sufficiently close to 1, i.e., as close to 1 as possible.

x	f(x)
0.5	1.50000
0.9	1.90000
0.99	1.99000
0.999	1.99900
0.9999	1.99990
0.99999	1.99999
…	…

From the table above we can see that as the values of x approach 1, the value of the function f(x) approach 2. In the table we have stopped at 0.99999, but if we take values of x even closer to 1, the corresponding values of f(x) will be even closer to 2. Here, as the values of x increase towards 1 the values of f(x) increase towards 2. This is symbolically written as:

It reads, “limit x tends to 1 plus f(x) is equal to 2”. It is to be noted that the ‘+’ sign signifies values of x greater than 1 and not “positive” values of x.

Let us now look at the following table

x	f(x)
1.5	2.50000
1.1	2.10000
1.01	2.01000
1.001	2.00100
1.0001	2.00010
1.00001	2.00001
…	…

Again from the above table we can see that as the values of x come closer and closer to 1, the corresponding values of f(x) come closer and closer to 2. In other words, as the values of x approach 1, the corresponding values of f(x) approach 2. The only difference is that here the values of x decrease towards 1 and the values of f(x) decrease towards 2. This is symbolically written as:

It reads, “limit x tends to 1 minus f(x) is equal to 2”. It is to be noted that the ‘-‘ sign signifies values of x less than 1 and not “negative” values of x.

In practice, the numerical difference between the value x=1 and a value of x sufficiently close to 1 (such as, x=1.0000001 or x=0.9999999) can me made as small as we please and hence can be neglected. Similarly, the numerical difference between the value f(x)=2 and a value of f(x) very close to 2 can be made as small as we please and hence be neglected if the value of x is taken sufficiently close to 1.

In general, when the value of f(x) cannot be determined for a particular value of x, say, x=a then there may exist a definite finite number b, such that the value of f(x) gradually tends to that finite number b when x tends to a. However we cannot say whether that finite number b will always exist or not. From this observation, the concept of limit has been developed by mathematicians.

Limit of a Variable

Let us consider a real variable x. Let a be a constant. Then by ‘x tends to a’ we mean x successively assumes values either greater than or less than a and the numerical difference between the assumed value of x, i.e., |x-a| becomes smaller and smaller. In this case, x becomes very close to a (but x≠a) and we say ‘x approaches a’.

If x approaches a assuming values greater than a then we say ‘x tends to a from the right side’ and we denote it by .

If x approaches a assuming values less than a then we say ‘x tends to a from the left side’ and we denote it by .

Limiting Value of a Function

We assume x to be a real variable, a is a real constant and f(x) is a single-valued function of x.

If x gradually approaches a assuming values which are greater than a and if the corresponding values of f(x) exist and these values gradually approach a finite constant , then $latexl_1$ is called the right hand limiting value of f(x) or the right hand limit of f(x) and it is denoted by,

Again, if x gradually approaches a assuming values which are less than a and if the corresponding values of f(x) exist and these values gradually approach a finite constant $latexl_2$, then $latexl_2$ is called the left hand limiting value of f(x) or the left hand limit of f(x) and it is denoted by,

When x approaches a assuming values either greater than or less than a and f(x) assumes finite values for every value of x and if those values of f(x) gradually approach a finite constant l, then l is called the limiting value of f(x). It is denoted by,

The limit of the function exists only if both (the right hand limit) and (the left hand limit) exist and , i.e., if

l does not exist if,

is indeterminate OR
is indeterminate OR
.

What do we mean by and ?

If a real variable x assumes positive values and increases without limit, taking up values larger than any large number one can imagine, then we say that the variable x tends to infinity in the positive direction and denote it by .

If a real variable x assumes negative values and increases numerically without limit, taking up values which are numerically larger than any large number one can imagine, then we say that the variable x tends to infinity in the negative direction and denote it by .

Some Important Limits

1. If n is a rational number, then

2. If n is a rational number, then

Points to Remember

does not mean that x takes positive values. It means that x approaches 0 assuming values greater than 0.
does not mean that x takes negative values. It means that x approaches 0 assuming values less than 0.
is called the right hand limit of f(x) at x=a and is called the left hand limit of f(x) at x=a.
exists if and both exists and
Limit of a function may not exist as and when the above conditions are not fulfilled. If not, the limit of the function at that point does not exist.

Questions and Answers

Question 1: Does exist? If so find its value?

Solution: Let

x	f(x)
1.9	3.9
1.99	3.99
1.999	3.999
1.9999	3.9999
…	…
2.1	4.1
2.01	.401
2.001	4.001
…	…

From the above table we can see that as x approaches the finite value 2 (assuming values either greater than or less than 2 and sufficiently close to 2) the values of f(x) gradually approach 4 and the difference between the values of f(x) and 4 can be made as small as we please. Hence, as , but x≠2 because f(x) is undefined at x=2.

Left hand limit =
Right hand limit=

Clearly both Right hand limit and left hand limit exist and are equal.

Therefore, exists and

Question 2: Evaluate

Solution: Let

x	f(x)
0.9
0.99
0.999
…	…
1.1
1.01
1.001
…	…

From the above table we can see that as x gradually approaches the finite value 1 from the left, assuming values less than 1, the value of f(x) keeps increasing and approaches a large number, as large as we can imagine.

Left hand limit =

Also, we can see that as x gradually approaches the finite value 1 from the right, assuming values greater than 1, the value of f(x) keeps increasing and approaches a large number, as large as we can imagine.

Right hand limit =

So left hand limit is equal to right hand limit.

Therefore,

NOTE: Here, x≠1 because at x=1, which is undefined.

Question 3: Evaluate

Solution : Let

x	f(x)
-0.1	-10
-0.01	-100
-0.001	-1000
…	…
0.1	10
0.01	100
0.001	1000
0.0001	10000
…	…

From the above table we can see that and

Clearly i.e., left hand limit and right hand limit exist, but they are not equal.

Therefore does not exists.

Question 4: Evaluate .

Solution : Let .

x	f(x)
-10	-1.02
-100	-1.0002
-1000	-1.000002
-10000	1.00000002
…	…

From the table we can see that as x increases numerically without limit, assuming negative values, the numerical difference between f(x) and the finite value 1 can be made as small as we please. Hence, as .