On-line Math 21

On-line Math 21

Example 3 Let
f(x) = ì
í
î
x2
if x ¹ 1
5
if x = 1
 .
Then

lim
x® 1 
 f(x) = 1,
even though f(1) = 5 .

Solution

Here the function is purposely defined to have the wrong value at 1. The limit

lim
x® 1 
 f(x)
is what f(1) ought to be.

This example stresses the predictability of values of the function. This function is bad in that it doesn't have the expected value at 1. The previous example had no value that f(1) should be; from one side it should be one thing, from the other side it should be another. But f has to have just one value.

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Copyright (c) 2000 by David L. Johnson.

File translated from TEX by TTH, version 2.61.
On 2 Oct 2000, 00:24.