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After learning the basic concept of limits, we can use limit properties to break complex limit calculations into simpler parts.
These properties are an important foundation in calculus because they let us calculate limits without always returning to formal definitions or value tables.
The simplest property is the limit of a constant function. If is a constant, then:
This means, the limit of a constant is the constant itself. This makes sense because the value of a constant does not change with respect to the variable .
For the identity function, the following holds:
When approaches , the value of function also approaches .
Suppose and where and are real numbers, then the following properties hold:
The limit of the sum or difference of two functions equals the sum or difference of the limits of each function:
This property allows us to break complex limits into simpler parts.
The limit of the product of two functions equals the product of the limits of each function:
A constant can be factored out from the limit sign:
The limit of the quotient of two functions equals the quotient of the limits of each function, provided the limit of the denominator is not zero:
with the condition .
The limit of a function raised to a power equals the power of the limit of the function:
where is a real number.
The limit of the root of a function equals the root of the limit of the function:
Important conditions:
Calculate .
Solution:
Using limit properties:
Calculate .
Solution:
Using division and multiplication properties:
Now we substitute the value :
In decimal form:
Calculate .
Solution:
Using the root property (since is even, we need to ensure the result inside the root is not negative):
Calculate the limit inside the root first:
Since , we can use the root property:
Calculate
Calculate
Calculate
Calculate
Calculate
Solution:
Using addition and multiplication by constant properties:
Substitute :
Solution:
Using the division property:
Solution:
Using the root property:
Solution:
Using the power property:
Solution:
Using the division property:
In decimal form:
