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Functions, Graphs, and Limits

Functions, Graphs, and Limits

At a Glance - Multiplying and Dividing Limits

Multiplication Property

As long as both and exist,

In words, the limit of a product is the product of the limits, as long as the limits involved exist.

Here's a few more examples of this:

  • Assume
    Then
  • Assume
    Then
  • Assume 
    Then

Sample Problem

Find

If we try to break this limit into pieces, we find

This is trouble, because

What do we find if we multiply ∞ by something (if we could do such a thing, which we can't because ∞ is not a number)? It would make reasonable sense to assume we'd get ∞ again.

What do we get if we multiply a number by 0? We get 0. What if we try to multiply ∞ by 0? That makes no sense at all.

It turns out that in this case we can't find the limit of the product via the product of the limits.
However, we can still find the original limit we were asked about.

Since

Division Property

If the limits of f and g exist and

,

then

Suppose and Then,

If , we can't use the rule we've been using to evaluate something like

.

If we try we find that

 

then we're out of luck.

Example 1

If

, and ,

what is

?


Example 2

Assume that  exists, , and .

What is ?


Example 3

Assume that  exists, , and .

What is ?


Example 4

Assume that  exists and .

What is ?


Exercise 1

Find the following limit, assuming that and .

  •  

Exercise 2

Find the following limit, assuming that and .


Exercise 3

Find the following limit, assuming that and .


Exercise 4

Find the following limit, assuming that and .


Exercise 5

Find the following limit, assuming that and .


Exercise 6

Assuming that  and ,find the following limit.


Exercise 7

Assuming that and , find the following limit.


Exercise 8

Assuming that  and ,find the following limit.


Exercise 9

Assuming that  and ,find the following limit.


Exercise 10

Assuming that  and ,find the following limit.


Exercise 11

Find  (we may assume this limit exists), given that

.


Exercise 12

Find  (we may assume this limit exists), given that

.


Exercise 13

Find  (we may assume this limit exists), given that


Exercise 14

Find  (we may assume this limit exists), given that

.


Exercise 15

Find  (we may assume this limit exists), given that

.


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