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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 and stays close to as approaches then we say that is the limit of as approaches . The value converges to is called the limit point.

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


Infinity In Limits

Limits can approach infinity:

If a limit approaching infinity is a real value, then as the variable gets larger and larger, the limit converges.

Limits can also be equal infinity:

On a graph, an infinite limit would appear to skyrocket or plummet to the top and bottom edges of the graph (asymptote).

Calculating Limits

Direct Substitution

If the function is continuous, limits can be evaluated by directly substituting the variable with the value it approaches. For example, because the cubic is continuous we have

In the limit below, direct substitution will give a false answer.

Graphing will show that is not continuous at , and that therefore does not exist.

Indeterminate Form

When a discontinuous function is improperly computed using direct substitution, one of the following peculiar results may be concluded:

The indeterminate forms do not provide enough information to identify the real limit.

The limits below are unequal. If they are directly substituted, they reach an indeterminate form and cannot be distinguished.

Discontinouities can be eliminated through algebraic manipulation.


L'Hopital's Rule

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