Limit Properties

So far we have worked with the idea of a limit and also defined limits carefully using epsilon and delta. This page deals with some properties of a limit under different conditions. The properties that will be discussed here, assuming and , are:

  1. ,

  2. ,

  3. ,

  4. .

The first property is saying when you take the limit of a sum, it is equal to the sum of the limits. The second one says when you multiply by a constant, L will also be multiplied by that constant. The third property says that the limit of a product is equal to the product of the limits. The last one says the limit of a quotient is equal to quotient of the limits.

Let's try to work some limit problems. Sometimes it may seem like the limit of a function is undefined since we are dividing by 0 in the denominator. If that happens we can work with the numerator to evaluate the limit. Take this limit for example:



It looks like we are dividing by 0 in the denominator as x approaches 7. However, let's factor the numerator:



As you can see we have common terms that cancel. After cancelling we are left with:



We can check our answer with a graph:

Let's try one more problem that's a little bit more difficult:


We have a zero in the denominator like before. We can try factoring, but its no good:

The trick here is to factor out a negative one from the denominator like this:


Now the terms cancel:

and we are left with the answer:

Once again checking our answer with a graph: