If a variable takes values which are more and more close to a finite number , then we say that approaches written as ).

  • If values of come closer to but are always greater than , then we say that approaches form right ().
  • If values of come closer to but are always less than , then approaches from left () .

The concept of a limit is essentially what separates the field of calculus, and analysis in general, from other fields of mathematics such as geometry or algebra.

The concept of a limit may apply to:

Examples

If , then can approach to '2' from two sides:

  • From right side: In notation we write means is coming closer to '2' from right i.e. it is more than '2'.
  • From left side: In notation we write mean is coming closer to 2 from left i.e. it is less than '2'.

Meaning of a limiting value

Let be function of . If the expression comes close to as approaches then we say that is the limit of as approaches .

In notation, it is written as .

Right Hand Limit

If approaches as approaches from the right, then is called as the right hand limit of .

Right hand limit can be expressed in two ways:-

Left Hand Limit

If approaches from the left, then is called the left hand limit of . The left hand limit can be expressed in two ways:-

Note that is a positive real number, that we let approach 0..

Existence of Limit

For existence of limit at

Illustrating the concept

If , then evaluate .

L.H.L. = i.e. is coming closer to 2 but it is less than '2'. So, observe the situation in table below:

1.9 0.1 3.9
1.99 0.01 3.99
1.999 0.001 3.999
Coming closer to 2 but less than 2 Coming closer to 4 but less than 4
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