## The delta epsilon proof

We have mentioned before that limits are what functions approach when the independent variable gets close to a certain value. Here is the problem-- as any obnoxious high school math teacher would put it-- please

We will succumb to the demands for once. The proof for a limit is called the "delta-epsilon" proof. The key point is to find a certain

We can write as: when |x-a|<delta, |y-L|<epsilon.

The key to the proof is to relate delta with epsilon. You are not seriously thinking of giving a table of epsilons and then making another table of deltas, and assuming they can continue forever, are you? If so, you have fallen into a common trap-- calculus, I tell you now, is half about continuing forever. The other half? Presenting infinity in a finite amount. That is the hard part. That is the interesting part.

For example, how do we prove that the limit of the function y=2x when x is 2 is 4? We need to find a delta, so that when x is within delta of two, y is within epsilon of 4. We need to relate delta to epsilon, so no matter what an epsilon you give me, I can find a corresponding delta. The solution here is that delta is equal to half of epsilon. (Take a moment to think about why) So if you require y to be within 2 of 4, I say x is within 1 of 2.

Note that most of the times, delta is smaller, and much smaller than epsilon. However, there is

1. The limit does not exist.

2. You need to revise your algebra. Click on Fun Algebra to proceed!

__proooove__it.We will succumb to the demands for once. The proof for a limit is called the "delta-epsilon" proof. The key point is to find a certain

*positive*number delta, so that when x is within delta of a number a, the value of the function f(x), must not diverge from the limit at x=a (suppose it's L) by another*positive*number epsilon.We can write as: when |x-a|<delta, |y-L|<epsilon.

The key to the proof is to relate delta with epsilon. You are not seriously thinking of giving a table of epsilons and then making another table of deltas, and assuming they can continue forever, are you? If so, you have fallen into a common trap-- calculus, I tell you now, is half about continuing forever. The other half? Presenting infinity in a finite amount. That is the hard part. That is the interesting part.

For example, how do we prove that the limit of the function y=2x when x is 2 is 4? We need to find a delta, so that when x is within delta of two, y is within epsilon of 4. We need to relate delta to epsilon, so no matter what an epsilon you give me, I can find a corresponding delta. The solution here is that delta is equal to half of epsilon. (Take a moment to think about why) So if you require y to be within 2 of 4, I say x is within 1 of 2.

Note that most of the times, delta is smaller, and much smaller than epsilon. However, there is

__one such delta__. If you can't find the certain delta, there are two possibilities:1. The limit does not exist.

2. You need to revise your algebra. Click on Fun Algebra to proceed!