Epsilon delta definition of limit examples

This section contains the formal limit of detection definition and limit of sequence definition. This is referred to by many as the epsilon-delta, relating to the alphabet of the Greek language. There are some informal ways to state a limit. 

When any function p = f(x) is given, along with the x-value and c,  then we can demonstrate it as a ‘limit of any function f, defined as x is approaching c, and it becomes the value of L.’

Suppose ‘y is defined as L’ in the same way that ‘x is defined as c’.

Suppose ‘y is approaching L’ in the same way that ‘x is approaching c’.

So, when ‘x is nearer to c’, in case of ‘y is nearer to L’.

Limit of detection Definition 

Let’s learn about the limit of detection definition: 

The limit of detection definition of the Epsilon-Delta, also termed the proof of Delta-Epsilon, is a formal definition that specifies a limit that is at a finite place with a value that is finite.

Epsilon delta definition of limit examples

Let’s learn about the epsilon delta definition of limit examples. 

The definition of the epsilon delta limit can be found with the following steps.

• We start by giving the definition of two different variables, epsilon (ε) and delta δ). 

• After this, we select a region of Epsilon lying the y-coordinate around the integer L.

• Along with this, we can make use of this region for the establishment of a delta zone nearer to the value of a on the x-coordinate. This is to make sure that all the values of x, which lie within the region, excluding a, will correspond to the y-values which lie within the region of Epsilon. 
• Then, we show that by using only the points in a small interval around a, we can get the function’s y-values, which are as near as we wish to the L value.
• We’ll define our limitations so that epsilon can represent any integer while delta can only be a certain value, thus, ascertaining that the region was exact.

Definition 1: a limit of any Function f

We can take I as an interval that is open and contains c, f as a function which is described as an I, with the only exception being c. As x is approaching c, then a limit of  function f(y) is L, is represented by the following:

This can be demonstrated as a,

lim f(y) = L

x→cf

This demonstrates that by providing any ϵ > 0, there is δ > 0, in a way that:

Each and every x ≠ c, suppose |x−c|<δ, then |f(y)−L|<ϵ.

Take note of the order in which it is presented. A p-tolerance is first specified in the formulation, and the limit will exist if we can find a q-tolerance that works.

This definition will be better understood if we take a look at an example. The explanation is lengthy, but it will guide you through all the needed stages in order to grasp the concepts.

Epsilon Delta Definition of Limit Examples

Example: Evaluating a limit using the definition

lim √x = 2

x – 4 

Solution:

Let’s attempt certain numerical tolerances before we utilise the formal definition. 

What happens if the y tolerance is 0.5 or equal to 0.5? 

How close to 4 must x be in order for y to be within 0.5 units of 2 i.e., 1.5y2.5? 

In this instance, we can take the following steps:

1.5 < y < 2.5 (1.2.5) 

1.5 < √x < 2.5 (1.2.6) 

1.52 < x < 2.52 (1.2.7) 

2.25 < x < 6.25 (1.2.8) 

So, what is the x tolerance that is desired? Remember that we’re looking for an asymmetric interval of x-values, precisely 4−δ < x < 4+δ. 

2.25 is 1.75 units away from 4, while 6.25 is 2.25 units away from 4. We require the smaller of these two lengths; thus, δ ≤ 1.75 is required.

Given the p-toleranceϵ = 0.5, we have found an x-tolerance, δ ≤ 1.75, in such a way, that whenever x is within δ units of 4, then y is within ϵ units of 2. 

Let’s try another value of ϵ.

What if the p-tolerance is 0.01. That is, ϵ = 0.01? 

How close to 4 does x have to be in order for y to be within 0.01 units of 2 (or 1.99 < y < 2.01)? 

Again, we just square these values to get

1.992 < x < 2.012

OR

3.9601 < x < 4.0401.(1.2.9)

What is the q-tolerance that you need? 

In this scenario, we need δ ≤ 0.0399, which is the shortest distance between 4 of the 2 bounds stated previously.

It’s worth noting that there appear to be two tolerances (below 4 of 0.0399 units and above 4 of 0.0401 units). However, we couldn’t use the larger value of 0.0401 for δ, since then the interval for x would be 3.9599 < x < 4.0401. This results in y-values of 1.98995 < y < 2.01 (which contains values NOT within 0.01 units of 2).

What we have so far: suppose ϵ = 0.5, then δ ≤ 1.75 and suppose ϵ = 0.01, then δ ≤ 0.0399. 

A pattern is not easy to see, so we switch to general ϵ to try to determine δ symbolically. We start by assuming y = x−−√ is within ϵ units of 2:

| y – 2 | < ϵ 

-ϵ < y – 2 <ϵ

-ϵ √x – 2 <ϵ

2 -ϵ √x – 2 <ϵ

(2- ϵ) 2 < x < 4 + 4ϵ + ϵ2

Conclusion

We sampled a few specific situations before dealing with the general issue in the preceding example, which was a little long. This isn’t done very often. The preceding example is likewise unsatisfactory in that 4–√=2. So, why strive so hard to show something that is so obvious? Many ϵ – δ proofs are lengthy and tough to complete. So, we learned about the limit of detection definition and the limit of sequence definition.