Chasing Infinity: Unveiling the Secrets of Limits (Part 1/3, CIA Submission)

The Tease: How Close is Close Enough?

Visualize creeping up on a buddy just for the purpose of surprising him. You're closing the gap, moving with deliberate caution until you're only a breath away from your friend. Alternatively, picture a basketball circling the rim, inching closer and closer to falling in. What is happening in those moments? You're approaching—but not quite reaching—something important. And that brings us to a fascinating question: what happens when we get infinitely close to something? In the world of mathematics, this has a name: limits.

The Basic Idea of a Limit

A limit is, in essence, about what value a function gets arbitrary close to when you approach a given number in the input. There is a caveat, though; it need not reach that number. Think of a graph that has a hole at a certain place. Even though the function isn’t defined there, the limit helps us make sense of what’s happening as we inch closer and closer to that point.

For example, consider the function: This function is undefined at this point because it is becoming 0 in the denominator. But as it drives near to 1 either way, it gets very very nearly 2. So the limit here is given by 2, although there is no function defined at. That, my friend, is the magic of limits: describing the behavior of functions in cases when a direct evaluation fails.

Why Limits Matter? 

Limits aren’t just a quirky mathematical tool—they’re the bedrock of calculus. Let’s explore some relatable examples to see why they’re so powerful.

1. The "Almost" Perfect Pizza

Imagine a pizza being cut into smaller and smaller slices. If you keep slicing infinitely, the size of each piece approaches zero. This idea underpins differentiation: breaking something into infinitely small parts to analyze change.

2. The Zoom Lens Analogy

Picture zooming in infinitely on a curve. As you magnify, the curve begins to look straighter and straighter, eventually appearing like a straight line. This is the concept behind the derivative, which uses limits to determine the slope of a curve at any given point.

Different Scenarios with Limits

Not all limits behave the same way. Let’s briefly touch on some possibilities:

1. Limits that Exist: Such situations arise when the function approaches a particular value in a very gradual manner.

2. Limits that Don't Exist: At times, functions display wild oscillation or discontinuity, rendering limit undefined.

3. Limits approaching Infinity: There are also cases where the function becomes infinite and not defined when the values of x take some values as they increase or decrease beyond all limits.

These scenarios showcase the versatility of limits in describing different types of behavior.

Limit Laws: The Rules of the Game

Limits aren’t a free-for-all; they follow specific rules that make them predictable and useful. Here are some key limit laws, in my own handwriting: 

These laws simplify complex calculations and give us tools to analyze even the trickiest functions.

(Here, f(x) and g(x) are the functions, l and m are the limit values, and a is the value which x tends to)

Left Hand Limit and Right Hand Limit

Sometimes, the behavior of a function differs as you approach a point from the left versus the right. This brings us to the concepts of left-hand limits (LHL) and right-hand limits (RHL):

• Left-Hand Limit: The value a function approaches as the input comes from values less than a certain point.

• Right Hand-Limit: The value a function approaches as the input comes from values greater than a certain point.

For a function to have a limit at a point, the left-hand limit and right-hand limit must be equal.

A practical example of analyzing the same is the flow of traffic at an intersection. Imagine cars approaching a traffic signal. From one side (the left-hand approach), cars slow down to stop. From the other side (the right-hand approach), cars are accelerating to cross the intersection. The traffic flow at the signal is continuous only if both sides match in speed and behavior.

Infinite Limits

Infinite limits describe what happens when a function grows without bound as the input approaches a particular value. These are important in understanding behaviors like vertical asymptotes or extremely rapid growth in real-world phenomena.

A practical example of relating with the same is the launch velocity of a rocket. Imagine a rocket accelerating as it approaches the speed needed to break free from Earth’s gravity. The closer it gets to this escape velocity, the faster it needs to accelerate, approaching infinity in theory if it were to reach the exact threshold.

Another example of an infinite limit is the above graph, wherein y = 1/(x)^2, The function shown likely has a vertical asymptote at x=1. This means as x approaches 1 from either side (($x\to {\mathrm{1^+ }}^{}$or $1^-$), the function’s value $y$ grows without bound, either positively or negatively. 

Concluding Thoughts

Limits are the very cornerstone of calculus, allowing a consideration of change, motion, and growth that would prove impossible in itself. From hacking a pizza into sly little slivers to closely zooming in on the curves, limits transform the view through which the mathematical world is o'er taken. They are more than a few digits mashed into a graph: they become a portal to infinity, to continuity, and to the very essence of change. So, next time you stalk a friend or try to cross the finish line, you should remember that--you are already playing with the concept of limits in wonder.

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Footnotes: The materials used in this blog, especially the textual definitions and references are taken from Thomas' Calculus textbook, Calculus textbook by Cengage and my class notes, along with images from various sources from Google.

And that is all for this blogpost on the interesting topic of "Limits" for my CIA Assignment submission, assigned by my Mathematics teacher, Dr. Ansa Mathew. Until next time folks, when I arrive with another anecdote of my adventures in Christ University!

- Divyabhanu Rana

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