LIMIT OF FUNCTIONS OF REAL VARIABLE

LIMIT OF FUNCTIONS OF REAL VARIABLE

CONTENT

INTRODUCTION     |   FUNCTION OF A REAL VARIABLE   |   TYPES OF FUNCTIONS   |   BASIC FUNCTIONS   |   LIMIT OF FUNCTIONS OF REAL VARIABLES

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

Let f be a function define for values of x near the point xo. Then the real number L is said to be the limit of f(x) as x approaches xo if, for each E>0 there exists a δ >0 such that  |f(x) – L| < E, whenever 0 < |x – x0|< δ  ; and we write 

Example

If f(x) = x2, show that

 

Solution

We have to show that given E > 0 there is a δ  > 0 ( in general, depending upon E) such that

  |x2 – 9|  < E, when 0 < | x – x0 | < δ . Let us choose δ    1 so that 0< | x-3 | < δ  1. Then we have  |x2 – 9| = |(x – 3) (x + 3)| = |x – 3||x + 3| < δ |x+ 3|   δ( |x – 3|+ 6) < 7δ .

Take δ  = min (1, E/2)  ,  the smaller of the two.

Then, we have  |x2 – 9|< E, whenever 0 <  |x – 3|< δ 1   

        We state without proof, the following easy prepositions.

Proposition:

If f and g are functions define for value of x near xo, and if


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