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

3.

4.

5.

6.

7.

## 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 .

Therefore .

Question 5: Evaluate

**Solutions:**

a.

=

=

=

= 7

b. Let

Now

Therefore

=

= 3 + 3 = 6

## Exercise

1. Show that

2. Evaluate . (Hint: factorize the numerator and the denominator)

3. Evaluate .

4. Show that .

5. Evaluate . [Hint: Multiply both the numerator and the denominator by . and rationalize]

6. A function f(x) defined as follows:

Examine whether exists or not.

7. Evaluate (Hint: Multiply both numerator and denominator by sin x. Put sin x = z)

8. Evaluate [Hint: Put ]

9. Evaluate [Hint: Put ]

10. Evaluate

shelly says

waooo i really love the way concept is given and elaborated with the help of solved examples..

upahar kc says

oh really is it helpful for solving calcules question.