LIMIT DEFINITION LIMIT

In this discussion, we confront the common understanding that the limit of 1/n as n goes to infinity equals zero. Seems straightforward, right?

Consider this: 1/2 + 1/2 equals 1, and 1/3 + 1/3 + 1/3 equals 1. In general, 1/n added n times equals 1. 0 + ... + 0 equals zero, not one leads us to the realization that the limit of 1/n as n approaches infinity, whatever it may be, is not exactly zero. Remember this perspective the next time someone claims limits like this are precisely zero.

The limit definition fundamental flaw can be visualized in this way: if we have a square with an area of -1, we can divide it into four equal parts, and repeat the process, each part having an area of -1/4 divided by four (16 pieces in total), essentially containing a scaled-down version of the original square in a fractal-like manner. However, the limit definition (an unproven statement) tells us that at some point, this exact but scaled-down version suddenly becomes zero and vanishes, no longer representing a square. The main challenge to accepting this idea is that we can put all these small pieces together and obtain the -1 unit square again. On the other hand, adding infinitely many zeros will result in zero and not -1. It is uncanny to believe that adding infinitely many zeros suddenly becomes a negative number.

The problems with the limit definition can usually be fixed by adding rules and clarifying the basic principles of continuity or the fifth postulate. However, these changes can sometimes lead to contradictions and inconsistencies in mathematics. Consistency is important; when we say 1 + 1 = 2, we want it to be true.

Additionally, these adjustments can be complicated, making us question their purpose. Some of these fixes are not real solutions; we may accept incorrect statements as facts, such as saying 0.999... = 1. Many people view this as a minor compromise for the sake of having a simpler limit definition. This is similar to believing the Earth is flat; it makes everyday life easier since we don’t have to think about the Earth’s curvature when we go shopping or walking in the park.

While this mindset is understandable, the mathematics we appreciate does not easily allow for compromises.

We want the limit definition to make sense and effectively address problems involving infinity. We don’t want to claim that some limit are indeterminate or sometimes imply that they are meaningless. We also want to avoid rules that do not consistently work, such as L'Hôpital's rule. Furthermore, we need the ability to distinguish between different sizes or kinds of infinity.

Limit Calculation

\[ \lim_{x \to \infty} \frac{x}{\sin(x) + x} \]

Limit Definition in Calculus

Comprehensive Definition of a Limit

In calculus, the limit of a function f(x) as x approaches a certain value a is a fundamental concept that describes the behavior of f(x) near a. Formally, we say:

lim_{x → a} f(x) = L

if for every ε (epsilon) greater than 0, there exists a δ (delta) greater than 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε.

Key Points:

Approaching a Value: The limit describes how f(x) behaves as x gets arbitrarily close to a, but not necessarily at a itself.
Epsilon-Delta Definition: The ε-δ definition provides a precise way to quantify what it means for f(x) to be "close" to L.
One-Sided Limits: Limits can be one-sided:
- Left-hand limit: lim_{x → a⁻} f(x)
- Right-hand limit: lim_{x → a⁺} f(x)
Limits at Infinity: The concept extends to limits as x approaches infinity, describing the behavior of f(x) as x grows without bound.
Existence of Limits: A limit exists if the values of f(x) approach the same number L from both sides of a.

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In the definition of limits, there is no option available to solve limits where the function or input value diverges to infinity. The videos on this page highlight the challenges associated with this limitation in a straightforward manner. To address this issue, we can recognize a new method that simplifies the process and closes the loophole in the conventional definition used by mathematicians.

The solution involves using a Chorological Progress Table, which is demonstrated in the the article on Riemann's Last Theorem (page 7) or videos related video to ∞=∞ (The Riemann Hypothesis and a New Math Tool ) . There are still a few steps to be addressed: incorporating different types of infinity into the limit definition and redefining it. Terence Tao's concept of ultimate limits, possibly inspired by Riemann's Last Theorem, has made strides in this area. However, progress has been slow in integrating various kinds of infinity, including anti-infinities and super-infinities.

In summary, the traditional definition of limits often feels inadequate, much like an ape gazing at a banana on a shiny tablet, confronted with the Transcendental Zeta Function and Super Symmetric equation. While the TZF clearly converges and has been proven through various methods, it raises questions about how this convergence differs from convergence in the limit definition, much like an ape observing a banana in the real world and wondering why it differs from a realistic photo on a tablet.