Why are Epsilon Delta proofs so hard?

I am a student, and I always find epsilon-delta proofs hard. The reason is the notation, and the number of things to keep in your head at the same time. It is always something like: For any epsilon, there exists a delta, such that it holds that …

What does Epsilon Delta prove?

The epsilon-delta definition of limits says that the limit of f(x) at x=c is L if for any ε>0 there’s a δ>0 such that if the distance of x from c is less than δ, then the distance of f(x) from L is less than ε. This is a formulation of the intuitive notion that we can get as close as we want to L.

How do you prove Delta Epsilon limits?

In general, to prove a limit using the ε \varepsilon ε- δ \delta δ technique, we must find an expression for δ \delta δ and then show that the desired inequalities hold. The expression for δ \delta δ is most often in terms of ε , \varepsilon, ε, though sometimes it is also a constant or a more complicated expression.

How does Epsilon Delta prove discontinuity?

To prove that f(x) is continuous in x=0, one would have to prove that for every ϵ>0, there exists a δ>0 such that when |x−0|=|x|<δ, then |f(x)−f(0)|=|x+1−0|=|x+1|<ϵ. To prove that f(x) is not continuous in x=0, one must show that there exists some ϵ>0 such that |x|<δ, but |x+1|≥ϵ.

What mathematicians did you find who were instrumental in developing the epsilon-delta definition of a limit?

In 1861, Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0. The modern notation of placing the arrow below the limit symbol is due to Hardy, which is introduced in his book A Course of Pure Mathematics in 1908.

Is Delta always less than epsilon?

Closed 3 years ago. In a delta-epsilon proof, you find a delta that you set to epsilon. This delta is less than or equal to epsilon.

Can Delta be equal to epsilon?

Therefore, since c must be equal to 4, then delta must be equal to epsilon divided by 5 (or any smaller positive value).

What is Epsilon continuity?

Among the sequence criterion, the epsilon-delta criterion is another way to define the continuity of functions. This criterion describes the feature of continuous functions, that sufficiently small changes of the argument cause arbitrarily small changes of the function value.

Can the limit of a function be infinity?

As a general rule, when you are taking a limit and the denominator equals zero, the limit will go to infinity or negative infinity (depending on the sign of the function).