Limit laws provide essential tools for analyzing how functions behave as their input approaches a specific value. These laws are particularly useful when dealing with combinations of functions, provided the individual limits exist. The Sum and Difference Laws state that the limit of the sum or difference of two functions equals the sum or difference of their respective limits:
The Product Law asserts that the limit of the product of two functions equals the product of their individual limits:
A practical example is a car rental cost function composed of three parts: a fixed cost, a mileage cost, and a discount factor, all depending on the number of rental days. To evaluate the limit of the total cost as rental days approach a specific value, the Product Law is applied first to separate the total cost into two functions. Then, the Sum Law is used to split the combined cost function into individual components. Each limit can then be calculated by direct substitution.
Limit laws are used to evaluate expressions involving combinations of functions, provided the individual limits exist.
According to the Sum or Difference Law, the limit of a sum or difference of two functions equals the sum or difference of their individual limits.
The Product Law states that the limit of the product of two functions equals the product of their individual limits.
These laws can be illustrated using a car rental cost model, where the total cost function includes a daily base cost, a mileage cost, and a discount factor—each expressed as a function of rental days.
As rental days approach a specific value, limit laws are applied to find the limit of the total cost function.
First, the Product Law is used to separate the limit of the total cost into the limit of the sum of the daily base and mileage cost functions, and the limit of the discount factor function—provided the individual limits exist at those points.
Next, the Sum Law is applied to split the sum into individual limits. This step is valid only if each limit exists. Each limit is then evaluated by substituting a specific number of rental days.
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