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asset101
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let an = [tex]\frac{(2n+1)}{(3n+7)}[/tex]. Prove direct from definition that an[tex]\rightarrow2/3[/tex].
Any help would be appreciated.
Cheers
Any help would be appreciated.
Cheers
That's the simplest way of finding the limit but it doesn't "prove from the definition".AUMathTutor said:Here's an idea... as long as n isn't zero, the following two things are the same:
(2n+1)/(3n+7) = [(2n)(1+1/2n)]/[(3n)(1+7/3n)]
What happens as the n's get bigger and bigger to the terms 1/2n and 7/3n?
"Prove direct from definition" is a scientific method used to validate a concept or idea by starting with its definition and logically deducing its properties and characteristics. It is a rigorous approach that relies on the fundamental principles of a concept to prove its validity.
"Prove direct from definition" is different from other methods of proof, such as induction or deduction, because it starts with the definition of the concept rather than making assumptions or generalizations. It is a more systematic and structured approach that relies on the exact meaning of a concept to prove its properties.
The steps involved in "prove direct from definition" include: 1) stating the definition of the concept, 2) identifying its fundamental principles, 3) using logic and reasoning to deduce its properties, and 4) providing evidence or examples to support the conclusions. These steps are repeated until the concept is fully validated.
"Prove direct from definition" has several benefits, including providing a thorough understanding of a concept, ensuring its validity, and avoiding errors or fallacies. It also allows for a clear and precise communication of ideas and can serve as a foundation for further research and experimentation.
"Prove direct from definition" is commonly used in scientific research to validate theories and concepts. For example, in mathematics, the Pythagorean theorem can be proven directly from its definition of the relationship between the sides of a right triangle. In physics, the laws of motion can be proven directly from the definition of force and mass.