On-line Math 21

1.2  Examples

1.2.1  Examples of basic limits

1. Let f(x) = 3x²+1 . Then
   

   lim x® 2

   f(x) = 13

   
   .
2. Let
   

   f(x) = ì í î 3x+1 if x ³ 1 x+2 if x < 1 .

   Then
   

   lim x® 1

   f(x)

   
   does not exist. The reason it doesn't exist is that, as you approach from the left ( x < 1 ), the values get towards 3, while from the right ( x > 1) , the values get towards 4.
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 . 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.

4. Let f(x) = sin(1/x) . Then
   

   lim x® 0

   f(x)

   
   doesn't exist. This one looks much worse than Example 2.. Here, as x® 0 , f(x) is wildly switching between -1 and 1.
5. Let
   

   f(x) =

   x3+1 x2-1

   
   . Then
   

   lim x® -1

   f(x) =

   -3 2

   
   .

1.2.2  Limits with infinity

1. 
   

   lim x® ¥

   Ö   x2+1   -1 x

   
   
2. 
   

   lim x® ¥

   x3+x2x-6 x3+5

   
   
3. 
   

   lim x® 1-

   1 x-1

   
   
4. 
   

   lim x® ¥

   x2+5x-6 x3+x+5

   
   

1.2.3  Examples of limits by the definition

Examples

1) Show that

lim x® 1

3x+5 = 8

.

2) Show that

lim x® 2

x2+3x = 10

.

3) Show that

lim x® 3

1 x

=

1 3

.

4) Find a number d sufficiently small so that the distance from f(x) = 2x²+3x-1 to 4 is less than 1/100 if |x-1| < d.

Example 1 Compute

lim x® 3

x2+2x-1 x3-3x2+2

.

Example 2 Compute

lim x® -1

Ö

x3+2x+7

.

To be fair, we really don't have the theory to claim this, but we can argue simply that, since

lim x® -1  x3+2x+7 = lim x® -1  æ è Ö   x3+2x+7   ö ø 2   = lim x® -1  Ö   x3+2x+7   lim x® -1  Ö   x3+2x+7   ,

and

lim x® -1

x3+2x+7 = (-1)3+2(-1)+7 = 4

, then

lim x® -1

Ö

x3+2x+7

= 2

.

Example 3 Compute

lim x® 9

x2-81 Öx-3

.

Example 4 Define

f(x): = ì ï í ï î   ___ Öx-4   , if x > 4 8-2x, if x < 4 .

Find

lim x® 4

f(x)

, if it exists.