- #1
fire sword
- 4
- 0
Homework Statement
http://store2.up-00.com/Sep12/JB498124.jpg [Broken]
2. The attempt at a solution
No attempts because i can't understand how to solve it
Last edited by a moderator:
Zondrina said:You could do this without any definitions for sin and cos.
micromass said:How can you possibly show that sin is continuous without defining sin?
Zondrina said:I suppose I see what you're saying here ^, but continuity of sin alone over ℝ would be sufficient enough to show that |sinx - sina| < ε
I would assume the continuity of sin was trivial though?
micromass said:The first question requires him to show continuity of sine. So I don't think he can assume that.
micromass said:The first question requires him to show continuity of sine. So I don't think he can assume that.
The epsilon-delta definition of a limit is a mathematical approach used to formally prove the existence of a limit. It involves selecting an arbitrary small value (epsilon) and finding a corresponding value (delta) within the domain of the function that will guarantee the output (y) will be within epsilon of the limit (L) for all inputs (x).
This definition is important because it allows us to prove the existence of a limit with mathematical rigor. It also helps us understand how a function behaves near a specific input and provides a way to quantify the concept of "close enough" when dealing with limits.
To prove a limit using the epsilon-delta definition, you need to start by selecting an arbitrary epsilon value. Then, using algebraic manipulation, you can express the distance between the limit and the output in terms of the distance between the input and the delta value. Finally, you can choose a delta value that satisfies the given conditions and proves the existence of the limit.
Some common mistakes when using this definition include not selecting an arbitrary epsilon value, not manipulating the expression correctly, and not choosing a delta value that satisfies the given conditions. It is also important to remember that this definition only proves the existence of a limit, not the actual value of the limit.
Yes, the epsilon-delta definition can be used for all types of functions, including continuous, discontinuous, and piecewise functions. However, the approach may differ depending on the type of function and the conditions given for the limit.