Limits, Mathematics

Limits

The concept of a limit is fundamental in calculus. Often, we are interested to know the behavior of f(x) as the independent variable x approaches some particular point 'a'. The question is, if we give values to x which are nearer and nearer to 'a', will the values of f(x) come nearer and nearer to any particular value? Suppose we define a function f(x) as:

                            f(x) = 2x

It can be seen that as we give values to x which are nearer and nearer to 0, then the value of f(x) also comes nearer and nearer to 0.

If x approaches a value 'a', f(x) approaches some number L, then we say that the limit of f(x) approaches L. This is symbolically written as

1669_limit.png        is to be read as 'x approaches a'.

Sometimes we may allow x to take values which are larger and larger, without any limit. This is symbolically written as  1954_limit1.png (read as 'x approaches infinity'). If f(x) approaches a limit L as  1967_limit2.png , then we write

1129_limit3.png

In some cases, it may so happen that as x approaches a value, the value of the function f(x) may become larger and larger without any limit. This is symbolically written as:

21_limit4.png

Example 

Suppose f(x) = 2x2 - 1

As x approaches value 1, f(x) approaches the value 1,

739_limit5.png

This is graphically represented below.

Figure 

1862_limit6.png

Posted Date: 9/13/2012 6:19:36 AM | Location : United States







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