Limits: Infinite Limits

To discuss infinite limits, let's investiagte the funtion f( x )= 5 x1 . Looking at the graph of this function shown here, you can see that as x 1 the value of f(x) decreases without bound and when x 1 + the value of f(x) increases without bound. A table of values will show the same behavior.

x

f( x )= 5 x1

x

f( x )= 5 x1

0.9

-50

1.00001

500,000

0.99

-500

1.0001

50,000

0.999

-5000

1.001

5000

0.9999

-50,000

1.01

500

0.99999

-500,000

1.1

50



A limit in which f(x) increases or decreases without bound as the value of x approaches an arbitrary number c is called an infinite limit.

This does not mean that a limit exists or that is a number. In fact the limit does not exist. The values of ± simply tell how the limit fails to exist because the values as x approaches c increase/decrease without bound.

Infinite limits are denoted by:

lim xa f( x )=


and is read "the limit of f(x) as x approaches a is infinity".

DEFINITION OF AN INFINITE LIMIT


Let f(x) be a function that can be defined on either side of a point a, and may or may not be defined at a:


lim xa f( x )=+       or      lim xa f( x )=


means as x approaches a, but not equal to a, the value of f(x) increase/decreases without bound.



The line at which the limit of a function increases or decreases without bound is called a vertical asymptote.

DEFINITION OF A VERTICAL ASYMPTOTE


The line x = a is a vertical asymptote of f(x) if one of the following is true:


lim xa f( x )=       lim x a f( x )=       lim x a + f( x )=

lim xa f( x )=       lim x a f( x )=       lim x a + f( x )=


Let's find the infinite limits in a couple examples.

Example 1: Find the lim x 4 + x x4 and lim x 4 x x4 , sketch the graph and define the vertical asymptote.

Step 1: Find the lim x 4 + x x4


Create a table of values for f(x) as x 4 + or justify the behavior of the values for f(x). Do not include x = 4.

x

x x4

4.1

41

4.01

401

4.001

4001

4.0001

40,001


As values for x get closer and closer to 4 but remain larger than 4, the denominator becomes a smaller and smaller positive number so the quotient becomes increasingly larger without bound.


lim x 4 + x x4 =

Step 2: Find the lim x 4 x x4


Create a table of values for f(x) as x 4 or justify the behavior of the values for f(x). Do not include x = 4.

x

x x4

3.9

-39

3.99

-399

3.999

-3999

3.9999

39,999


As values for x get closer and closer to 4 but remain smaller than 4, the denominator becomes a smaller and smaller negative number so the quotient becomes increasingly large negative number without bound.


lim x 4 x x4 =

Step 3: Sketch the graph.


Step 4: Define the vertical asymptote.


Because lim x 4 f( x )= ; lim x 4 + f( x )= , the line x=4 is a vertical asymptote.

Example 2: Find the lim x5 x1 x+5 if it exists. If it does not exist explain why.

Step 1: Find the lim x 5 + x1 x+5


Create a table of values for f(x) as x5 or justify the behavior of the values for f(x). Do not include x = -5.

x

x1 x+5

-4.9

-59

-4.99

-599

-4.999

-5999

-4.9999

-59,999


As values for x get closer and closer to -5 but remain larger than -5, the denominator becomes a smaller and smaller positive number while the numerator approaches -6 so the quotient becomes increasingly negative without bound.


lim x 5 + x1 x+5 =

Step 2: Find the lim x 5 x1 x+5


Create a table of values for f(x) as x 5 or justify the behavior of the values for f(x). Do not include x = -5.

x

x1 x+5

-5.1

61

-5.01

601

-5.001

6001

-5.0001

60,001


As values for x get closer and closer to -5 but remain smaller than -5, the denominator becomes a smaller and smaller negative number while the numerator approaches -6 so the quotient becomes an increasingly larger positive number without bound.


lim x 5 x1 x+5 =

Step 3: Sketch the graph.


Step 4: Determine if the function has a limit.


Because lim x 5 f( x )= ; lim x 5 + f( x )= , the line x=5 is a vertical asymptote. Because the limits increase without bound, no limit exists. Remember, an infinite limit is not a limit but merely states how the limit fails.





Related Links:
Math
algebra
Limits: Limit Laws
General Differentiation Rules
Calculus Topics


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