Epsilon-Delta

 

The Formal Definition of the Limit


e-d Definition of a Limit

Let f(x) be a function and L be a number we say that

         

if for any choice of e, the d team can respond with a positive number d so that with a "perfect calculator" the d team will win.  That is for any 
0 < e ,
there is a 0 < d such that for all x with 

          0 < | x - a| < d 

we have

          0 < | f(x) - L| < e 





Example:   

Show that if 

        f(x) = 7x 

then 

       

Solution:  

Let 

        0 < e

Scratch Work:

 


we need to find a d such that

        14 - e  <  f(x)  <  14 + e

for all 

        2 - d  <  x  <  2 + d

or equivalently

        14 - e   <   7x  <  14 + e

or after dividing by 7,

        2 - e/7  <  x  <  2 + e/7

 


If we choose 

        d  =  e/7

then

        2 - e/7  <  x  <  2 + e/7

implies that

        14 - e  <  7x  <  14 + e

so that

        14 - e  <  f(x)  <  14 + e

which proves that the limit is14.



Exercise  

Prove that

A)  If 

        f(x)  =  3 - 5x

then

         lim as x goes to 4 = -17

B)  if 

        f(x)  =  mx + b
 

is a line then

       


A Proof of a Limit that Does Not Exist


Example:  Prove that the function

       

does not have a limit at x = 2

Solution:  

Let 

        e  =  .5

then for any chosen d, chose 

        m  = min(d/2,0.01) 

so that

        f(2 - m)  =  (2 - m)2 - 1  =  3 - 4m + m2   <  3.1 

and

        f(2 + m)  =  (2 + m) + 3  =  5 + m  >  4.9.

Now for any L either

        |3.1 - L|  > 0.5    or |   4.9 - L|  >  0.5

hence the limit does not exist.  Below is the graph.

       

Notice that on the left hand side the limit approaches 3 and on the right hand side it approaches 5.





Exercise:
 

Prove that if 
f(x) = { 3x - 5 for x < 1
2x + 2 for x > 1


then the limit does not exist.



Limits and Graphs

If f(x) is a function, then the limit as x approaches c is L if the y coordinates of the left hand side from x = c of the graph and the right hand side of the graph both approach L.  Graphically, we can get a good guess of what the limit is by putting the function into a graphing calculator and checking to see if the left and right agree and the y coordinate is likely to be the limit.  Once we have a guess of what the limit is, we can use the epsilon-delta definition to attempt to prove that what the calculator indicated is indeed the limit.  For most functions and values of c, the conjecture that the calculator investigation produces will turn out to be correct; however, occasionally the calculator will produce misleading results.

Other Sites About Limits

Karl's Calculus

Visual Calculus

Ohio State Calculus

Weisstein's World of Mathematics

Dr. Sloanes Calculus 

 

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