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Definition of Limit

Main Concept

The precise definition of a limit states that:

 

Let  f  be a function defined on an open interval containing c (except possibly at c) and let L be a real number.

 

Define the limit of  f  at c to be L, or write

 

 limxcfx = L 

if the following statement is true:


For any ϵ > 0  there is a δ > 0 such that whenever

     0 < xc < &delta;

then also

 f&lpar;x&rpar;L < &epsilon; .

 

Suppose you want to prove that a certain function has a limit. What exactly needs to be determined?

An input range in which there is a corresponding output. (A positive δ so that xc < &delta;   f&lpar;x&rpar;L < &epsilon;.)

Example 1

Prove:

limx25 x  1 &equals; 9

Note: c  &equals; 2 &comma;  L &equals;9&comma;  fx &equals; 5 x1 .

Remember you are trying to prove that:

For all &epsilon; &gt; 0, there exists a  &delta; &gt;0 such that:

if 0 <  x  2 < &delta;  then 5 x1 9 < &epsilon;.

Step1: Determine what to choose for &delta;&period;

f&lpar;x&rpar;L  < &epsilon;

5 x1 9  < &epsilon;

Substitute all values into f&lpar;x&rpar;L  < &epsilon;.

5 x10 < &epsilon;

 

5 x2 < &epsilon;

 x2 < &epsilon;5

The relation has been simplified to the form xc < &delta;, if you choose &delta;&equals;&epsilon;5.

 

Step 2: Assume  xc < &delta;, and use that relation to prove that f&lpar;x&rpar;L  < ε.

xc<&delta;

x2<&epsilon;5

Substitute values for c and &delta;.

5x2<&epsilon;

5 x10 < &epsilon;

5 x1 9 < &epsilon;

f&lpar;x&rpar;L  < &epsilon;

Follow the instructions, using different functions  f, values of c, ϵ and δ to observe graphically why the proof works.

 

1. Choose a function:

2. Choose a value for c:

c  =

3. Ask for an &epsilon;:

&varepsilon; =

4. Try to choose &delta; small enough so that xc < &delta; implies fxL < &epsilon;.T. If the blue strip is a river, and the purple strip is a bridge, then the function (green) must only cross the river where the bridge is!

 

&delta; =

5. If it's not possible to choose such a &delta;, the function fx does not have a limit at the point c !

 

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