(ε, δ)-definition of limit
Ok, imagine you have a toy car and you want it to go really fast. You have a race track that goes around in a circle, and you want your car to go around the circle as fast as possible.
Now, you remember that you learned about limits before. A limit is like a rule that tells you how close you can get to something without actually reaching it. So in this case, a limit would tell you how close your car can get to going as fast as possible without actually going that fast.
But how do you figure out this limit? That's where the (ε, δ)-definition of limit comes in. It's like a special way of thinking about limits.
Ok, let's break it down. The ε in the (ε, δ)-definition stands for "epsilon". Imagine epsilon is a number that tells you how close you want your car to get to going as fast as possible. So if you pick a really small epsilon, that means you want your car to get really close to the fastest speed.
Now, the δ in the (ε, δ)-definition stands for "delta". Think of delta as another number that tells you how close your car needs to be to the circle for it to go as fast as possible. If you pick a small delta, that means your car needs to stay really close to the circle.
So the (ε, δ)-definition of limit says that for any small epsilon you pick, there will be a small delta that tells you how close your car needs to be to the circle. And if your car stays within that delta, it will get really close to going as fast as possible (within that epsilon limit).
So let's try an example. If you say you want your car to get within 10 miles per hour of the fastest speed, that would be your epsilon. Then, there would be a delta that tells you how close your car needs to be to the circle, let's say within 5 feet. If your car stays within 5 feet of the circle, it will go really close to the fastest speed within that 10 mile per hour limit.
So the (ε, δ)-definition of limit is like a special way to think about how close you can get to something without actually reaching it. It uses epsilon and delta to show that you can get really close to the limit by being within a certain range. It's like a special type of race track for your toy car, where you can go really close to the edge without falling off.
Now, you remember that you learned about limits before. A limit is like a rule that tells you how close you can get to something without actually reaching it. So in this case, a limit would tell you how close your car can get to going as fast as possible without actually going that fast.
But how do you figure out this limit? That's where the (ε, δ)-definition of limit comes in. It's like a special way of thinking about limits.
Ok, let's break it down. The ε in the (ε, δ)-definition stands for "epsilon". Imagine epsilon is a number that tells you how close you want your car to get to going as fast as possible. So if you pick a really small epsilon, that means you want your car to get really close to the fastest speed.
Now, the δ in the (ε, δ)-definition stands for "delta". Think of delta as another number that tells you how close your car needs to be to the circle for it to go as fast as possible. If you pick a small delta, that means your car needs to stay really close to the circle.
So the (ε, δ)-definition of limit says that for any small epsilon you pick, there will be a small delta that tells you how close your car needs to be to the circle. And if your car stays within that delta, it will get really close to going as fast as possible (within that epsilon limit).
So let's try an example. If you say you want your car to get within 10 miles per hour of the fastest speed, that would be your epsilon. Then, there would be a delta that tells you how close your car needs to be to the circle, let's say within 5 feet. If your car stays within 5 feet of the circle, it will go really close to the fastest speed within that 10 mile per hour limit.
So the (ε, δ)-definition of limit is like a special way to think about how close you can get to something without actually reaching it. It uses epsilon and delta to show that you can get really close to the limit by being within a certain range. It's like a special type of race track for your toy car, where you can go really close to the edge without falling off.
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