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Functions, Graphs, and Limits

Functions, Graphs, and Limits

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.

  • Assume
  • Assume
  • Assume 

Using the result from the previous example,


Sample Problem


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.



Division Property

If the limits of f and g exist and


we find

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 are out of luck.

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