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 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.