Limits and Continuity: Pre-Calculus
Before we move on to calculus and its vast real-world applications, we must first understand the concept Limits and Continuity. These are called pre-calculus and without a thorough understanding of these two ideas, there cannot be a proper intuition about calculus. Calculus is a tool for understanding change and limits and continuity are the foundations that make this possible.
What exactly is a Function in the Real World?
Students mostly hate it in the beginning (few may be exceptions). To get a feel of what it really is, understand that it is simply a rule that tells us how one quantity depends on another. If one thing changes and another thing changes because of it, we have a function. We see speed as a function of time (rate of change of distance), electric current as a function of time (rate of change of charge), temperature and heat can be functions of space and time, strain in a material can be a function of length, width, or height. Every physical quantity we study—motion, heat, electricity, stress, economics—can be written as a function. When we write a system as a function (specifically a polynomial), we move from guessing to modelling (weather, population, finance), controlling machines (cars, drones, EV motors), designing engineering systems, creating algorithms (AI, robotics, economics).
Limit of a Function and Continuity around a Point
A good way to start would be the common example of instantaneous velocity – Imagine tracking how far a car moves in a tiny slice of time. If you check distance every 1 second, you get average speed. If you check every 0.1 sec, you get a more accurate speed. If you check every 0.0000001 sec, you get almost the exact speed at that moment. That limiting value is instantaneous velocity. For e.g. Speed shown of the speedometer of car. There can be no concept of speed at 0 sec because if time does not move, there is no distance covered!
In the illustration we clearly see what we mean by infinitesimally small changes and also why we not talk about ‘0’ in Limits. Limit means boundary. So the limit of a function describes what value the function approaches as the input gets very close to a specific point (time in the above illustration) i.e. the value of f(x) as ‘x’ approaches a specific point. The function need not actually reach that value. The thing to remember is that arrival of ‘x’ to the value must be smooth (not in breaks and jumps). A limit is said to exist when the function behaves in a smooth and stable manner around that point. But why is it called a “Limit”? Because it represents the value that a function “approaches” or is “limited to” as the input gets arbitrarily close to a certain point. You may or may not actually reach the point but the limit tells you where the function is heading. For a limit to exist, the graph must approach the same value from both sides of the point and its behaviour near the point must be smooth.
Things in nature we see around us show smooth transition. Change is continuous but in a smooth manner. What we won’t find in the real world functions are jumps, wild oscillations, different behaviour around the point (left and right). When we say that “Limit Exists Around a Point”, it means values just smaller and just larger than the point but not the exact point. Therefore, as input ‘x’ gets closer and closer to a certain value ‘a’, the value of function ‘f(x)’ settles down to a single number. There are no jumps, spikes and unpredictability.
So here comes Continuity, the second part after Limits and the stronger requirement for calculus. This stable and consistent behaviour of a function at and near a point is exactly what we mean by Continuity around that point i.e. tiny changes in input produce tiny changes in output. Neighbouring points in space or time have nearly the same value. Say if a small change in the value of an input, which can be time, position, temperature, force, current causes a smooth change in the response of physical system, then that function is continuous. Quantities like speed, temperature, current, voltage, temperature generally depend on time. Electric and magnetic fields, stress and pressure generally depend on space.
We know that speed of a car cannot jump directly from 0 km/hr to 100 km/hr. It changes smoothly. Voltage doesn’t jump from 200 V to 240 V suddenly, it rises continuously. Sudden jump means infinite power which is physically impossible to exist in a real system. Electric current rises smoothly and does not jump from 5 A to 50 A unless there is a fault (which breaks the real system). Heat also spreads gradually and the temperature in a rod does not jump suddenly at a single point. Stress is redistributed smoothly through the material as infinite stress does not occur in real solids. Thus, in real systems everything changes gradually, smoothly, without jumps. This smoothness is called continuity. If real systems had jumps, it would imply infinite force or speed or current.
So continuity depends on limits. A function is continuous at a point if the value at the point as well as from its left and right match.
Why Limits and Continuity are Pre-Calculus
Interesting fact is that there can be no calculus without the concept of Limits and Continuity! Every physical quantity keeps changing with time or space. Calculus is the tool that studies how fast and how much something changes and in what way. Nearly every concept in calculus is about change, be it speed, acceleration, force, growth, decay.
But at a single point, nothing changes. A point is static. And this is the key idea. Calculus is about measuring change and to measure change (which is always relative) we need the ideas of ‘limits’, ‘approaching’, ‘continuity’. Calculus studies change by examining how a quantity behaves as we get closer and closer to a point in time, space, or input. For example, speed at an instant, current at a moment, temperature change in a moment, stress at a point etc. That is exactly what limits describe. Without limits, calculus collapses completely because instantaneous speed cannot be defined, slopes at a point cannot be measured, derivatives and integrals cannot exist.
Every major law of physics and engineering relies on limit-based thinking. We will understand it better with examples (like the example of instantaneous speed discussed before):
Electric Current at a Point – Electric current is how fast charge flows. Current at a point is how many electrons cross that point in one tiny instant of time i.e. a small amount of charge based on time it takes to pass. If you measure charge flowing every 1 second, that’s rough. If you measure in even smaller time slices, you get the exact current at that point. No time duration means no movement of charge. That’s why the concept of infinitesimals.
Heat Conduction – How fast heat moves from a hotter region to a colder region. It depends on temperature difference between two points and distance between them. If you measure temperature change over a small distance, you get an approximate heat flow. If you shrink the distance to almost zero (but not zero), you find the exact heat flow at that point inside the material. Heat does not jump; it flows point by point, through very infinitely thin layers.
Stress and Strain in Materials – How much a material stretches compared to its original length i.e. small change in length as a ratio of total length. We measure change not over the whole rod, but at an infinitely small point inside it because real materials don’t stretch evenly everywhere. The redistribution of stress is continuous but not uniform.
Irony of nature is that if we were to measure all the above changes at an exact instant, it would not be possible because at a single instant, nothing actually changes as time is frozen. We can’t calculate a rate from zero change in zero time. To measure dynamic quantities like velocity (change in position), current (change in charge), heat flow (change in temperature) and strain (change in length), we need change andapproximation is done using very small changes. A rate of change needs two things, a “before” and an “after”. Because there is no “before” and “after” at a single instant, so you cannot compute the rate. So mathematics gives a method to measure change in small intervals and shrink the intervals till they become infinitesimally small. Smaller the change we can detect, greater the precision and control can achieve. It is like zooming into the picture infinite times to see the behaviour at an instant. Thus, Limits and infinitesimals give physics the ability to describe reality with perfect precision through measurements occurring over finite (non-zero) intervals. Wonderful concept!
Endgame
Limits define well-defined approach. Continuity ensures nature behaves smoothly there. Calculus uses these ideas to define change, motion, and accumulation. The practical applications of Limits i.e. infinitesimal changes and Continuity i.e. smoothness of change are explored deeply in calculus since calculus itself is impossible without them.
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