June 2022

Between both high school and college, I figure I must’ve spent at least 500 hours studying calculus. Today, I use exactly zero calculus in my daily life. No derivatives. No integrals. No differential equations.

So, a reasonable question to ask might be, “Did I just waste over 500 hours of my life on a useless subject?” As an optimizer, it’s these types of questions - ponderings about where in life I’ve lost valuable time - that often surface.

But the short answer is no. I don’t see the many hours I spent studying calculus as wasted time. Now, I could go on at length about how calculus was invaluable as an intellectual challenge, or how it allowed me to view problems from a unique vantage point, or how my calculus learnings would later complement my understanding of financial derivatives along with a handful of other subjects. And certainly, all these benefits would be true. But, interestingly, the most valuable learning from my Calculus studies came from the application of a simple concept called “Limits”.

From Wikipedia:

*“In mathematics, a limit is the value that a function approaches as the input approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.”*

Put simply, limits are tools to understand functions.

Take a basic function like y = x. One way to understand it is by graphing it. Start with x = 0, identify the corresponding output (here, y would also equal zero), plott the point on a graph, and repeat this process over and over incrementing the x-value by 1 unit with each iteration. Quickly, you’ll see that this function takes the shape of a simple line with a slope of 1.

But let’s say you’re not necessarily concerned with all the minutiae of the function. You don’t care about each specific point. You just want to quickly understand the function at a high level. This is where “Thinking in Limits” is useful.

Take another simple function like y = 1 / x. What happens to this function when its input (x) takes on extreme values? Well, as x approaches infinity, the denominator becomes massive and pulls the function’s value closer and closer to 0. Here, we would say that the limit of the function as x approaches infinity is zero. Using simple abstraction (with limits), we can predict how a function should behave.

What they don’t teach you in calculus is that “Thinking in Limits” can and should be applied to one’s personal life as well. How so?

Well, our lives can be thought of as complex functions. On the x-axis is time. On the y-axis is productive or “good” output. Just as we use limits to help understand the nature of mathematical functions, so too can we use limits to understand the likely direction of our future.

You see, our actions, our behaviors, and our thoughts can be thought of as inputs to our personal life function. If this is true, then a powerful question to ask is, “What does my life function look like when the limit of my inputs (my actions, behaviors, and thoughts) approach infinity?”

The reason why “thinking in limits” is so powerful is because it forces you to face this consequential truth that, ultimately, it’s your defaults that define you.

Roughly five years ago, I started consuming podcasts intensely. Not out of obligation but out of a genuine curiosity and hunger for knowledge. Topics ranging from personal finance to philosophy to health science to macroeconomics. Listening to podcasts became my default activity. Over the course of the last five years, I estimate that on average I’ve spent ~2.5 hours every single day consuming podcast content.

Here’s some basic math on my consumption:

[5 years] x [365 days / yr] x [2.5 hours / day] = 4,562.5 hours

That’s right, from age twenty to twenty five, I digested almost 5,000 hours worth of podcasting content. Now, as big as this number might seem, what happens when I project this rate of consumption further into the future? Well, by age thirty, I’ll have consumed nearly 10,000 hours worth of podcasts, by age fifty 25,000 hours, and by age eighty (if I ever live to see the age) I’ll have digested over 50,000 hours worth of podcasts…

Like I said, “Thinking in Limits” forces one to acknowledge that it’s your defaults that ultimately define you.

Most people are myopic with their behavior. They’re thinking about the next day, the next week, maybe even the next year. But most people aren’t truly “thinking in limits”. They’re not asking themselves, “What would my life look like if I made this same exact decision every single day for the rest of my life?”

When you start framing your daily decisions in this way, it changes how you view seemingly meaningless activities. Because now, engaging in a 30-minute social media death-scroll is no longer just a quick break. If that activity is a daily habit then what that activity really is is a forfeiting of thousands of hours of your future life. (Spending 30 minutes on an activity everyday for only 1 year works out to 10,950 minutes.)

Now, don’t get the wrong idea here. The point is not to condemn leisurely activity. Watching TV. Scrolling through social media. Playing video games. I honestly think there is a good and appropriate time for these activities. But the unfortunate reality is that most people (myself included) rarely realize the long-term, consequential impact that these daily defaults have on their future. “Thinking in Limits” helps with this.

As the renowned American writer, historian, and philosopher William Durant once stated:

“We are what we repeatedly do. Excellence, then, is not an act but a habit.”