Jun 9, 2024

Sky(e) is the limit

Using e as an introduction to limits

#math 2 min read

Limits are an interesting concept. Being new to this myself, I was trying to visualise this to understand the core concept.

What are limits?

Suppose, you're a equation of variable $x$ . And you're trying to plug in numbers into it. And you find that the larger you plug in, the closer the equation seems to get to a certain number. That number is the limit.

Limit Visualisation

It's like the more you try to resist your urge to eat an ice-cream, the closer you get to actually eating one. (That's my visualisation.)

e as a limit

Let's take an expression:

\begin{matrix} (1 + \frac{1}{n})^{n} \end{matrix}

This looks awfully similar to the equation of :Compound Interest; because it kinda is.

But what happens if you start chugging in numbers to $n$ ?

\begin{aligned} n = 1, (1 + \frac{1}{n})^{n} = 2 \\ n = 10, (1 + \frac{1}{n})^{n} = 2.593 \dots \\ n = 100, (1 + \frac{1}{n})^{n} = 2.704 \dots \\ n = 1000, (1 + \frac{1}{n})^{n} = 2.7169 \dots \\ n = 10000, (1 + \frac{1}{n})^{n} = 2.7181 \dots \end{aligned}

... we see that it is getting closer and closer to the :Euler's Constant!

Really fascinating! I suppose all the smug businessmen will leave their shops and start doing Calculus after seeing this.

To actually see it happening

I made a Python script to plug in values of x in increments, and I could really see it happening. This helped a lot in empirically understanding the concept.

from sympy import * # This package allows you to do complex mathematical equations

iteration = 0 # where x = iterations

while True:
	iteration += 1000 # increments

	n = symbols("n")
	expression = (1+1/n)**n # (1+1/x)^x

	number = limit(expression, n, iteration)

	print(f"When n = {iteration} -> {float(form)}") # Prints result

Video Result

So yea, there you go. There are fun stuff happening in Maths (and all around the world), and once we get exposed to it, there's no going back.

Happy Practicing!

:x Euler's Constant

It is a mathematical constant 2.718281828... named after mathematician Leonhard Euler. Something like the $π$

This comes in handy in Logarithm as well.

:x Compound Interest

\begin{matrix} A = P (1 + \frac{I}{N})^{N T} \end{matrix}

Where:

A is the Compound Amount
P is the Principal Amount
I is the Interest Rate
N is the no. of times interest is compounded per year
T is the no. of years.

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Sky(e) is the limit

Using e as an introduction to limits

What are limits? #

e as a limit #

To actually see it happening #

:x Euler's Constant #