Some of the content of this guide was modeled after a guide originally created by Openstax and has been adapted for the GPRC Learning Commons in September 2021. The graphs are generated using Desmos. This work is licensed under a Creative Commons BY 4.0 International License.
Described in this section are the properties of limits, assuming that c is a constant and the following limits exist:
| Property | Description |
|---|---|
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The limit of a sum is equivalent to the sum of the limits |
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The limit of a difference is equivalent to the difference of the limits |
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The limit of a constant times a function is equivalent to the constant times the limit of the function |
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The limit of a product is equivalent to the product of the limits |
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The limit of a quotient is equivalent to the quotient of the limits
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The limit of a function to the power of n is equivalent to finding the limit of the function and applying the power of n to the result
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The limit of a function to the root of n is equivalent to finding the limit of the function and applying the root of n to the result
or
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Table 1: Properties of Limits
The limit of sums is used when there are a combination of functions using addition. The two separate parts of the function can be solved for the limit and those limits can be summed separately.
For the function below, the two parts of the function can be separated and solved individually before recombining the results.
Apply the basic limit laws and calculate the limit:
The limit of differences is used when there are a combination of functions using subtraction. The two separate parts of the function can be solved for the limit and those limits can be subtracted separately.
For the function below, the two parts of the function can be separated and solved individually before subtracting the results.
As described, if a function consists of a constant as a multiplier, the constant can be removed and the limit of the function can be solved with the multiplier applied afterwards.
The limit of products is used when there are a combination of functions using multiplication. The two separate parts of the function can be solved for the limit and those limits can be multiplied separately.
The limit of quotients is used when there are a combination of functions using division. The two separate parts of the function can be solved for the limit and those limits can be divided separately, if the limit of the denominator does not equal zero.
