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Functions of Two Variables: Limit of a function- 1 Video Lecture | Mathematics for Competitive Exams
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FAQs on Functions of Two Variables: Limit of a function- 1 Video Lecture - Mathematics for Competitive Exams
| 1. What is the definition of the limit of a function of two variables? |  |
Ans. The limit of a function of two variables is a fundamental concept in calculus that describes the behavior of the function as the input values approach a particular point. It represents the value that the function approaches as the input values get arbitrarily close to the specified point.
| 2. How is the limit of a function of two variables evaluated? | |
Ans. To evaluate the limit of a function of two variables, we need to consider the behavior of the function along different paths approaching the specified point. If the function approaches the same value regardless of the path taken, then the limit exists and is equal to that common value. However, if the function approaches different values along different paths, the limit does not exist.
| 3. Can the limit of a function of two variables exist even if the function is not defined at the specified point? | |
Ans. Yes, the limit of a function of two variables can exist even if the function is not defined at the specified point. The limit only depends on the behavior of the function as the input values approach the point, not the actual value of the function at that point. So, even if the function has a hole or a removable discontinuity at the specified point, the limit can still exist.
| 4. What are the properties of limits for functions of two variables? | |
Ans. The properties of limits for functions of two variables include:
- The limit of a sum or difference of two functions is the sum or difference of their limits.
- The limit of a product of two functions is the product of their limits.
- The limit of a constant times a function is equal to the constant times the limit of the function.
- The limit of a quotient of two functions is the quotient of their limits, provided the denominator function does not approach zero.
- The limit of a composition of two functions is the composition of their limits, provided the limit of the inner function exists.
| 5. How do we determine if the limit of a function of two variables does not exist? | |
Ans. The limit of a function of two variables does not exist if the function approaches different values as the input values approach the specified point along different paths. This can happen when there is a jump discontinuity, an essential discontinuity, or when the function oscillates or diverges along different paths. Additionally, if the function approaches infinity or negative infinity as the input values approach the point, the limit is said to be infinite and does not exist in the traditional sense.
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Nov 22, 2024 Last updated |
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