Limits : Never Quite Reaching Absolute (Part-1)

 Greetings !

Folks, have you ever wondered how you can never achieve absoluteness in this world? Like speed of light cannot be attained by anything with mass... Or how you can never actually touch things? Yes, it is true, the electrons around your entire body have repulsion and so they don't allow anything to actually touch you. Quite interesting right? It is surprisingly similar to limit functions not being able to reach to the value they are trying to, let alone going beyond it...

I always found these science facts really interesting and i notice similarities to them in mathematic concepts too now! I got reminded of limits once when i was trying to zoom into a really high quality pic and it wouldn't stop to zoom, i just couldn't reach the end of it. I felt very limited at that moment (pun intended).

So this is how I related to limits, i find this topic very amusing. Let's talk some more about the history of limits

Greek Philosophy and Mathematics

Zeno, a pre-Socratic philosopher left us with a series of paradoxes which highlight the difficulty in understanding how an infinite series of steps could lead to a finite outcome. Which directly raises questions towards nature of infinity and and the very concept of limits. (Around about 585-470 BCE).

Archimedes, a renowned physicist and mathematician, came up with "Methods of exhaustion" using which he could find the area or volume using polygons and gradually increasing the number of sides of them to obtain more accurate values. (Around about 230 BCE).

The concepts we study now emerged much later but these ideas played a major role in the study of limits.

"Achilles and the Tortoise"

Formal Definition of a Limit (ε-δ Definition)

The formal definition of a limit uses the Greek letters epsilon (ε) and delta (δ) to express the idea of closeness:

Limit of a function at a point: The limit of a function f(x) as x approaches a is L, denoted by

lim_{x→a} f(x) = L 

if for every ε > 0, there exists a δ > 0 such that 

0 < |x - a| < δ ⇒ |f(x) - L| < ε   

 Basically as 'x' gets nearer and nearer to a value 'a', the function f(x) approaches L. It can approach the value from both sides, hence there are two types of limits LHL (left hand limit) and RHL (right hand limit).

Calculation of a limit

Discussing this in a practical manner, for lim_{x→a} f(x) = L to exist,

Two things are a must :-

1) LHL and RHL exist

2) LHL =  RHL

Limits laws and techniques

Here are some fundamental limit laws that simplify the evaluation of limits:

1. Sum/Difference Rule: lim_{x→a} [f(x) ± g(x)] = lim_{x→a} f(x) ± lim_{x→a} g(x)

2. Product Rule: lim_{x→a} [f(x) * g(x)] = lim_{x→a} f(x) * lim_{x→a} g(x)

3. Quotient Rule: lim_{x→a} [f(x) / g(x)] = [lim_{x→a} f(x)] / [lim_{x→a} g(x)], provided lim_{x→a} g(x) ≠ 0

4. Constant Multiple Rule: lim_{x→a} [c * f(x)] = c * lim_{x→a} f(x), where c is a constant

Techniques for Evaluating limits

• Direct Substitution: If the function is continuous at the point, you can simply substitute the value of a into the function.

• Factoring: If there are indeterminate forms (0/0, 0×∞,∞/∞, ∞ −∞, ∞0, 00, 1∞.), factoring the numerator and denominator can sometimes help you cancel out common factors and evaluate the limit.

• L'Hôpital's Rule: This powerful rule applies to indeterminate forms (all mentioned above). By using this rule you can easily just differentiate numerator and denominator separately to remove the indeterminate form and solve the equation. 

Applications of limits

• Physics:

The concept of limits is binded and etched into this world. It has numerous applications like in physics, when we are talking about anything instantaneous it means when time is tending to zero. There are concepts like instantaneous velocity or instantaneous acceleration where we consider average of the value in smaller and smaller intervals.

• Computer Graphics

Limits play a vital role in rendering smooth curves and surfaces on computer screens. There are many mathematical functions to define shapes and determine points in these curves.

• Economics

Marginal revenue, economic growth models, risk management, financial knowledge, and many other things require the use of limits at one place or the other.

It doesn't end at just this, There is a lot more to it.

Conclusion

Limits are the building blocks of calculus. They provide the base for understanding concepts like derivatives and integrals, which are fundamental tools in various fields of sciences and engineering. Without a solid grasp of limits, it would be impossible to develop the powerful techniques of calculus that are used to model and solve complex problems.

In my next blog, we'll delve into the concept of continuity. We'll know about how limits play a crucial role in understanding the seamless flow of the unbroken functions and how the behavior of a function near a point is actually connected to its continuity at that point. Stay tuned!