Limits: Introduction and One-Sided Limits

The limit of a function f(x) as x approaches some arbitrary value a is denoted by:

lim xa f( x )=L


and is read "the limit of f(x) as x approaches a equals L". So as the value of x approaches the value of a, the value of f(x) approaches L. It is important to note that the limit does not include where x = a but only the values close to and on either side of a.

Take the function f( x )= x+2 x1 as it approaches 1. The table lists the values of f(x) near x = 0.

x

f( x )= x+2 x1

x

f( x )= x+2 x1

-0.1

-1.72

0.0001

-2.0003

-0.5

-1.8571

0.001

-2.003003

-0.01

-0.1.9702

0.01

-2.030303

-0.001

-1.997003

0.05

-2.157895

-0.0001

-1.9997

0.1

-2.333333

The table shows that as x approaches 0 from either the left or the right, the value of f(x) approaches -2. From this we can guesstimate that the limit of f( x )= x+2 x1 as x approaches 0 is -2:

lim x0 ( x+2 ) x1 =2




While the limit of the function f( x )= x+2 x1 seems to approach -2 as x approaches 0 from either the left or the right, some function have only one-sided limits. The following notation is used to denoted left-hand and right-hand limits.

lim x a f( x ) lim x a + f( x )

Left-hand limitRight-hand limit

Let's take a look at some limits of the function graphed below.

The limit as x approaches 1 from the left, lim x 1 f( x ) , is 3 while the limit as x approaches 1 from the right, lim x 1 + f( x ) , is 1. Since the left-hand and right-hand limit as x approaches 1 are different, the limit as x approaches 1 does not exist.

Now' let's look at the limits as x approaches 2. The limit as x approaches 2 from the left is 0.5, and the limit as x approaches 2 from the right is 0.5. Therefore the limit as x approaches 1 from either direction is 0.5.

Note that the when x = 2 the value of the function is 4. Thus the limit is not concerned about the value when x = 2 but only the value as x appraoches 2.

Limits as x1

Left-hand Limit:    lim x 1 f( x )=3

Right-hand Limit: lim x 1 + f( x )=1

Overall Limit:      lim x1 f( x ) = DNE

Limits as x2

Left-hand Limit:   lim x 2 f( x )=0.5

Right-hand Limit: lim x 2 + f( x )=0.5

Overall Limit:     lim x2 f( x ) = 0.5





Let's find the limits in a couple examples.

Example 1: Sketch a graph and create a table to determine the limit of the function as x2 .

lim x2 x1 +3 x

Step 1: Graph the function.


Step 2: Create a table of values close to and on either side of 2.


x

f( x )= x1 +3 x

x

f( x )= x1 +3 x

1.8

2.1635

2.01

1.9925311

1.9

2.0782

2.05

1.9632659

1.95

2.0382

2.1

1.9280042

1.99

2.0075314

2.2

1.861566


The value as x approaches 2 from both the left and the right approaches 2.


lim x 2 f( x )=2 ; lim x 2 + f( x )=2

Step 3: Guesstimate the limit

Since the limit from both the left and the right are the same, then the overall limit as x approaches 2 is 2.


lim x2 f( x )=2 .

Example 2: Sketch a graph and create a table to determine the limit of the function as x1 .

lim x1 x 6 +2 x 8 +1

Step 1: Graph the function.


Step 2: Create a table for values close to and on either side of 1.


x

f( x )= x1 +3 x

x

f( x )= x1 +3 x

0.9

1.7696

1.0001

1.4997

0.99

1.5298

1.001

1.4969

0.999

1.5029

1.01

1.4969

0.9999

1.5003

1.1

1.1997628


The value as x approaches 1 from both the left and the right approaches 1.5.


lim x 1 f( x )=1.5 ; lim x 1 + f( x )=1.5

Step 3: Guesstimate the limit

Since the limit from both the left and the right are the same, then the overall limit as x approaches 1 is 1.5.


lim x1 f( x )=1.5 .

Example 3: For the function whose graph is given state the value of each quantity if it exists. If it does not exist, explain why.

lim x 3 f( x )       lim x 3 + f( x )       lim x3 f( x )       f( 3 )       lim x 1 f( x )       lim x 1 + f( x )       lim x1 f( x )

Graph of Function


Step 1: Evaluate the limits as x approaches 3.


lim x 3 f( x ) - As x approaches 3 from the left, the value of f(x) approaches 2.


lim x 3 + f( x ) - As 3 approaches 3 from the right, the value of f(x) approaches 0.


lim x3 f( x ) - Since the value of f(x) as x approaches 3 from the left does not equal the value as x approaches 3 from the right, this limit does not exist.

Step 2: Evaluate f(3).

The value of y when x is 3 is -1.


f(3) = -1


Notice that the limits of f(x) as x1 from the left or right may not be related to the value when x = 1.

Step 3: Evaluate the limits as x approaches 1.


lim x 1 f( x ) - As x approaches 1 from the left, the value of f(x) approaches 2.


lim x 1 + f( x ) - As x approaches 1 from the right, the value of f(x) approaches 2.


lim x3 f( x ) - Since the value of f(x) as x approaches 1 from the left equals the value as x approaches 1 from the right, this limit is also 2.





Related Links:
Math
algebra
Limits: Infinite Limits
Limits: Limit Laws
Calculus Topics


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