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jmed
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Homework Statement
evaluate lim2x^2 as x approaches 3 using formal definition (epsilon and delta) of limit
What have you attempted?jmed said:Homework Statement
evaluate lim2x^2 as x approaches 3 using formal definition (epsilon and delta) of limit
Homework Equations
The Attempt at a Solution
jmed said:epsilon> 0 and (|x-1|) <delta...delta equals ??
x^2 - 1 has nothing to do with this problem.jmed said:delta= a fraction of epsilon so that it is less than x^2 - 1?
jmed said:limit is 18...I am just confused with this whole process...
The limit of 2x^2 as x approaches 3 using epsilon-delta definition is 18. This means that as x gets closer and closer to 3, the value of 2x^2 will approach 18.
The epsilon-delta definition of a limit is a mathematical method used to precisely define the concept of a limit. It states that the limit of a function f(x) as x approaches a is L if, for any positive number ε, there exists a positive number δ such that if the distance between x and a is less than δ, then the distance between f(x) and L is less than ε.
To find the limit of a function using the epsilon-delta definition, we first choose a value for ε (the desired distance between f(x) and L). Then, we use algebraic manipulation to find a corresponding value for δ (the distance between x and a) that satisfies the definition. This value of δ will ensure that the distance between f(x) and L is less than ε for all x within that distance of a.
The use of epsilon-delta definition is important because it provides a rigorous and precise way to define and calculate limits, especially in more complicated functions where other methods may fail. It also allows for a deeper understanding of the concept of a limit and its relationship to a function's behavior near a particular point.
Yes, the epsilon-delta definition can be used to prove the existence of a limit. If we can find a value for δ that satisfies the definition for any given value of ε, then we can conclude that the limit exists. However, it is important to note that this method does not necessarily provide a numerical value for the limit, but rather proves its existence.