Most high school students dread the subject of calculus. This fear may be why even many smart students stay away not only from higher mathematics but also STEM subjects and go into Commerce, Psychology, Economics, etc. I believe that the main reason for this calculus-phobia is the way it is presented, and I would like to offer an alternative perspective.

First, it should not even be called by a strange name like calculus (which is also used to describe undesirable accumulation on teeth). A more appropriate name is “Algebra and Trigonometry using a new concept”. If students are good in Algebra and Trig, they would be enticed by this description to know the concept. The concept is that of “limit”. It is a shame that the concept was presented in my first calculus course in such an abstract, overwhelming way, using the so-called “epsilon-delta definition” that I was ready to quit and enrol in a statistics class. Subsequently, I concluded that this was the most fascinating concept in my entire learning experience of mathematics.

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I was appreciative of the need for an idea like limit long before I studied calculus. As an avid follower of statistics of Indian cricket players in test matches it was clear to me that if a player had a batting average of say, 50 per innings and had a couple of bad matches with lower scores his average would drop below 50. Even if he continued to score 50 in every inning thereafter, his average would never get back to 50.

There were riddles like if you cover half of the remaining distance in each step in approaching a destination how many steps you must take to reach there. The answer is that you will never get there but can approach as close as you want by increasing the number of steps. Later, in my academic career I saw a similar issue in GPA calculation. If you are a straight-A student but somehow end up with a B in any course, you will never get back to a 4.00 average again even if you get an A in every course afterwards but can only approach that. I often wondered how the speedometer of a car worked. Speed is distance divided by the time of travel.

One could measure its average speed over smaller and smaller time intervals with accurate odometers and clocks; more and more accurately by reducing the time interval but not quite the speed at any given instant. Later, I learned that the speedometer reading is linked directly to the speed of rotation of the wheels by electromagnetic means! Even in the spiritual world, the concept of limit helps us to understand our destiny. Our gurus teach us that one can reach God through meditation; the longer one meditates the closer one gets to God. We never really reach God but can experience Him in closer and closer proximity through longer and more intense meditation.

I was finally intrigued and satisfied when I learned the concept of limit. If a variable x approaches some number a, and correspondingly some other variable related to x, called a function of x, denoted by f(x) approaches L, we say, as x approaches a, the limit of f(x) is equal to L. As an example, consider the function f(x) = (X^2 –1)/(x-1). At x =1, it is undefined because it reduces to 0/0. However, if we plug in values of x closer and closer to 1 (such as 1.1, 1.05, 1.01, 1.005 etc.) in the expression for f(x) we will find that it approaches the number 2. We say that as x approaches 1, the limit of f(x) is 2.

It can also happen that as x approaches a certain value, a function f(x) can continue to get larger and larger and not approach any finite number. In that case we say that the limit of f(x) as x approaches a is infinite. Over the years it sank into me that the concept of limit is not only a powerful concept in Calculus it is really the only new concept. Whether one calculates the derivative of a function, derives the rules of differentiations or finds the integral of a function, understanding the concept of limit is essential.

The rest is simply algebra and trigonometry which can be admittedly somewhat tedious at times. Take differential calculus as an example; there, the central concept is that the derivative of a function for a certain value of x is the slope of tangent to the graph of that function at that x. The concept of limit allows us to find slope of the tangent from the slope of a chord line connecting two points on the graph by bringing the two points closer and closer – not a difficult idea. Once one knows this slope, one can determine if the function is increasing, decreasing, going through a peak or valley or just remains constant.

If we further explore to see if the slope itself is increasing or decreasing with x, we can predict more details about the variation of the graph such as its concavity. Suppose we want to determine the volume inside a flower vase. As a first try, we may fill it up with tiny steel balls, then pour the balls out and count them. The volume of the vase is approximately equal to the number of balls times the volume of each ball.

Everyone would recognize that it is not exact because there are spaces between the balls. As a next step in refinement, we may want to fill the vase with sand. If we want an exact measurement, we will fill the vase with water and then measure the volume of water in a measuring cylinder. In the language of limit, we would say that the volume of the vase is the limit of the sum of volumes of all the balls (or grains) when their number becomes larger and larger. This is the key concept in integral calculus in a nutshell.

Calculus will be a lot more interesting if we first present its practical applications which everyone will understand and then go into algebraic computations. The real beauty of the concept of limit (and more generally, that of calculus) is that it allows us to express so many complicated aspects of functional relationships between different quantities in a compact and elegant way.

Just imagine the difficulty and awkwardness in stating most of the laws of physics if calculus was not invented. It is no wonder that Newton had to invent calculus to clearly express his laws of physics. They say that the sky’s the limit if one wants to learn mathematics. Using mathematical jargon, we can say that as the amount of time devoted to learning math approaches infinity, the limit of math knowledge is infinite!

(The writer, a physicist who worked in industry and academia, is a Bengali settled in America.)