Limits Properties

PROPERTIES OF LIMITS

First we will assume that Limits Properties1http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif and  Limits Properties2http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif exist and that c is any constant.  Then,

  1. Limits Properties3http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif

In other words we can “factor” a multiplicative constant out of a limit.

  1. Limits Properties4

So to take the limit of a sum or difference all we need to do is take the limit of the individual parts and then put them back together with the appropriate sign.  This is also not limited to two functions.  This fact will work no matter how many functions we’ve got separated by “+” or “-”.

  1. Limits Properties5

We take the limits of products in the same way that we can take the limit of sums or differences.  Just take the limit of the pieces and then put them back together.  Also, as with sums or differences, this fact is not limited to just two functions.

  1. Limits Properties6http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif

As noted in the statement we only need to worry about the limit in the denominator being zero when we do the limit of a quotient.  If it were zero we would end up with a division by zero error and we need to avoid that.

  1. Limits Properties7http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif

In this property n can be any real number (positive, negative, integer, fraction, irrational, zero, etc.).  In the case that n is an integer this rule can be thought of as an extended case of 3.

For example consider the case of n = 2.

Limits Properties8http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif

The same can be done for any integer n.

  1. Limits Properties9

This is just a special case of the previous example.

Limits Properties10http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif

  1. Limits Properties11http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif

In other words, the limit of a constant is just the constant.  You should be able to convince yourself of this by drawing the graph of Limits Properties12http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif.

  1. Limits Properties13http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif

As with the last one you should be able to convince yourself of this by drawing the graph of Limits Properties14http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif.

  1. Limits Properties15http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif

This is really just a special case of property 5 using Limits Properties16http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties_files/empty.gif.

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