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pivoxa15
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Can the epsilon associated with f(x) be a function of x?
i.e epsilon = delta * (x-2)^2 valid?
i.e epsilon = delta * (x-2)^2 valid?
learningphysics said:No. It shouldn't be function of x.
Epsilon and delta are variables used in an epsilon-delta proof to define the notion of a limit. Epsilon represents a small positive number, and delta represents a small positive distance. The goal of an epsilon-delta proof is to show that as the distance between the input and the limit point (represented by delta) approaches zero, the output of the function (represented by epsilon) also approaches zero.
Epsilon-delta proofs are used to formally prove the existence of a limit in a mathematical function. They provide a rigorous and precise way to define the behavior of a function near a certain point, and are an essential tool in the field of calculus.
In general, an epsilon-delta proof consists of three main parts: the setup, the proof, and the conclusion. The setup involves defining the variables epsilon and delta, and stating the limit definition. The proof involves manipulating the inequalities involving epsilon and delta to prove that the limit holds. The conclusion states that the limit has been proven.
Epsilon-delta proofs can be challenging to understand at first, as they require a solid understanding of calculus and mathematical reasoning. However, with practice and guidance, they can become easier to comprehend. It is important to break down each step and understand the logic behind it in order to fully grasp the concept.
Yes, epsilon-delta proofs can be used for all functions, as long as the function is defined and continuous at the given point. However, the complexity of the proof may vary depending on the function and the point in question.