I think your teacher starts out with a given ε > 0, and then figures out what δ needs to be.anthonyg said:Thanks, the thing is I don't know where to start. I mean my teacher always starts out with given ε > 0 such that δ... I honestly just don't know where to go either.
The statement is trying to prove that if the absolute value of a function f(x) is less than or equal to a constant B for all values of x except for x=0, then the limit of the product of x and f(x) as x approaches 0 is equal to 0.
This statement can be proven using the definition of a limit and the properties of absolute value and inequalities.
Having a limit as x approaches 0 means that the function approaches a specific value as x gets closer and closer to 0. In other words, the function's output becomes more and more predictable as x gets closer to 0.
The absolute value is significant because it ensures that the function's output is always positive, even if the input is negative. This allows us to make conclusions about the behavior of the function without worrying about its sign.
This statement is related to the concept of continuity because it shows that the function is continuous at x=0, meaning that the limit of the function at that point exists and is equal to the function's value at that point. This is an important property in the study of calculus and real analysis.