Introduction
Around the turn of the 18th century, a great dispute shook the world of mathematics. On one side stood the English physicist Isaac Newton, and on the other, the German mathematician Gottfried Wilhelm Leibniz. The prize? Bragging rights to the title of Father of Calculus. Their rivalry sparked one of the longest mathematical controversies in history, lasting for decades. But today, we recognise them both as co-founders of this powerful branch of mathematics.
What Are Limits?
Calculus comes in different flavours such as integrals and derivatives, but at its core lies the concept of limits. Understanding limits is key to mastering the subject.
If you’ve ever been on a long road trip with kids, you’ve probably heard the classic question: Are we there yet? Limits in calculus answer a similar question—what happens as we get closer and closer to a destination? But instead of a road trip, we’re dealing with numbers and functions. Limits tell us what value a function is approaching, even if it never quite reaches it.
How Limits Work
Imagine you’re driving to point A, which is 100 km away. Every hour, you check Google Maps and see how close you are:
- After 1 hour → 50 km left
- After 2 hours → 25 km left
- After 3 hours → 12.5 km left
- After 4 hours → 6.25 km left
Each hour, you’re getting closer and closer to 100 km travelled. But what happens as you approach your destination? This is exactly what limits in calculus help us understand—what value we are approaching, even if we never actually reach it.
In math terms, if f(x) represents the total km travelled, the limit describes what happens as x (time) gets closer and closer to the final destination (100 km).
Mathematical Notation:
This means that as x (time) approaches infinity, the function f(x) (km travelled) approaches 100 km—even if we don’t explicitly say what happens when x is exactly ∞.
Why Do We Need Limits in Calculus?
For starters, without limits, there wouldn't be calculus. They provide a foundation for derivatives and integrals. Limits also help us to make predictions like estimating stock prices, modelling population growth or understanding physics.
Conclusion
Limits are more than just a mathematical formality, they help us understand real-world behaviours. Whether it’s tracking a road trip’s progress or analyzing the speed of a cheetah, limits give us a way to describe change in a precise manner. But this is just the beginning!
In future posts, we’ll dive into derivatives—the next big step in calculus—where we go from asking "Are we there yet?" to "How fast are we getting there?"
If you’re curious to dive deeper into limits and see more examples, here are some amazing resources that helped me:
Khan Academy - Limits Intro - A structured breakdown of limits with practice problems.
The Organic Chemistry Tutor(YouTube) - Clear, step-by-step explanations on calculus topics.
3Blue1Brown – The Essence of Calculus (YouTube) - An intuitive approach to understanding calculus concepts.
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