wat is the significance of the epsilon value in analysis?

Answer 1

Epsilon is typically a small value close to zero, but not zero. While epsilon is arbitrarily chosen when it comes to limits, delta depends on epsilon. So it's kind of like the independent variable when you are proving limits, but chosen to be very small. The limit definition says that for all epsilon>0, there exists a delta>0 such that when |x-c|

Answer 2

Typically, epsilon is used as part of an "epsilon-delta" proof. This is a proof of any of several conditions (continuity, differentiability, convergence, etc.) from a definition along the lines of
"... if for any epsilon > 0 there exists a delta > 0 such that for all [expression] < delta, [expression] < epsilon."
There's also a related form typically using N instead of delta, and with the "[expression] < delta" condition replaced by "n > N".

Example: a sequence (x_n) converges to value x if, for any epsilon > 0, there exists an N > 0 such that for all n > N, ||x_n - x| | < epsilon.

Example: a function f(x) is continuous at x_0 if, for any epsilon > 0, there exists a delta > 0 such that for all x with ||x - x_0|| < delta, ||f(x) - f(x_0)|| < epsilon.

Epsilon represents an "error" or "variance" amount. The idea behind these definitions is that no matter how small you make the error tolerance, we can find a suitable region in which the error is smaller than that amount. So this implies that we can get arbitarily close to the function value by taking close enough points, or that we can get arbitarily close to the limiting value of the sequence if we go far enough into it.