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The expansion of limit term refers to the process of calculating the limit of a function as the independent variable approaches a specific value. It involves evaluating the function at values closer and closer to the specified value to determine the limit.
The expansion of limit term is important because it allows us to determine the behavior of a function at a specific point, even if the function is not defined at that point. It is also a fundamental concept in calculus and is used in many real-world applications.
The basic rules for expanding limit terms include the sum rule, product rule, and quotient rule. The sum rule states that the limit of the sum of two functions is equal to the sum of their limits. The product rule states that the limit of a product of two functions is equal to the product of their limits. The quotient rule states that the limit of a quotient of two functions is equal to the quotient of their limits.
No, the expansion of limit term can only be used for continuous functions. This means that the function must be defined and have a smooth, unbroken graph with no jumps or holes. Functions that are not continuous may have different limits from the left and right sides of the specified value, making it impossible to calculate the limit using the expansion method.
One common mistake when expanding limit terms is to assume that the limit exists without actually calculating it. It is important to evaluate the function at values closer to the specified value to determine if the limit truly exists. Another mistake is to use the expansion method for functions that are not continuous, as mentioned in the previous question. It is also important to note that the expansion of limit term does not always work for more complex functions and may require the use of other techniques, such as L'Hôpital's rule.
