Introductory calculus courses often give students the following rules-of-thumb when finding limits of ratios of functions:
For example, this implies \(\lim\limits_{x\to\infty} \frac{\exp(x)}{x^{2} + 7x + 42} = \infty\) or \(\lim\limits_{x\to\infty} \frac{\log(x)}{x^{4}+x-12} = 0\).
However, courses tend not to emphasize just how ridiculous the factorial function is in comparison. Recall that the factorial of a non-negative integer \(n\) is defined such that
\[\begin{equation} n! = n\cdot(n-1)\cdots 2\cdot 1, \end{equation}\]where \(0! = 1\) by convention. So, say \(n = 4\). Then \(n! = 4\cdot 3\cdot 2\cdot 1 = 24\).
Factorials are absolute monstrosities for even moderately-sized \(n\). To appreciate just how quickly factorials grow, let’s consider looking back \(n!\) seconds in the past. Given a number of seconds, the following how_long_ago function finds its equivalent formulation in years to weeks all the way to seconds.
how_long_ago <- function(seconds) {
# given seconds, returns years, weeks, days, hours, mins, and secs
if (seconds < 0) stop("seconds must be non-negative")
spm <- 60 # seconds per minute
sph <- 60 * spm # per hour
spd <- 24 * sph # per day
spw <- 7 * spd # per week
spy <- 52.1775 * spw # per year
s <- seconds
years <- floor(s / spy); s <- s %% spy
weeks <- floor(s / spw); s <- s %% spw
days <- floor(s / spd); s <- s %% spd
hours <- floor(s / sph); s <- s %% sph
mins <- floor(s / spm); s <- s %% spm
cat(years, "years,", weeks, "weeks,", days, "days,",
hours, "hours,", mins, "minutes, and", s, "seconds.")
}
You were likely reading my brief introduction or clicking on the link to this page \(4!\) seconds ago.
how_long_ago(factorial(5))
## 0 years, 0 weeks, 0 days, 0 hours, 2 minutes, and 0 seconds.
how_long_ago(factorial(6))
## 0 years, 0 weeks, 0 days, 0 hours, 12 minutes, and 0 seconds.
how_long_ago(factorial(7))
## 0 years, 0 weeks, 0 days, 1 hours, 24 minutes, and 0 seconds.
\(7!\) seconds ago you were probably waking up, heading to work, eating some food, or, if you’re like me, procrastinating on the computer.
how_long_ago(factorial(8))
## 0 years, 0 weeks, 0 days, 11 hours, 12 minutes, and 0 seconds.
\(8!\) seconds ago is almost a half-day in the past. I was getting ready for bed (grad students live exciting lives).
how_long_ago(factorial(9))
## 0 years, 0 weeks, 4 days, 4 hours, 48 minutes, and 0 seconds.
\(9!\) seconds ago from Wednesday at noon (when I published this post) is Saturday morning. Do they still show cartoons on Saturday morning?
how_long_ago(factorial(10))
## 0 years, 6 weeks, 0 days, 0 hours, 0 minutes, and 0 seconds.
\(10!\) seconds ago from February 4, 2015 (the publish date of this post) yields December 24, 2014. You may have been celebrating Christmas Eve or the last night of Hannukah.
how_long_ago(factorial(11))
## 1 years, 13 weeks, 5 days, 18 hours, 10 minutes, and 48 seconds.
From February 4, 2015, \(11!\) seconds ago it was October 31, 2013. Happy Halloween!
how_long_ago(factorial(12))
## 15 years, 9 weeks, 2 days, 8 hours, 42 minutes, and 0 seconds.
\(12!\) seconds ago from now we entered the final month of the 20th century: December 1, 1999. With 2000 AD just around the corner, the world was collectively freaking out about the Y2K bug.
how_long_ago(factorial(13))
## 197 years, 17 weeks, 0 days, 5 hours, 27 minutes, and 36 seconds.
Holy history Batman! \(13!\) seconds ago it was the year of 1817. In the United States, President James Monroe had recently bought Florida from Spain and was in the midst of his nation-wide good-will tour promoting national unity and ushering a relatively young America into the Era of Good Feelings.
how_long_ago(factorial(14))
## 2762 years, 29 weeks, 5 days, 5 hours, 9 minutes, and 36 seconds.
\(14!\) seconds ago it was the 8th century BC. Ancient Greece transitioned from the Dark Ages into its Archaic period and held the first-ever Olympic games in 776 BC.
how_long_ago(factorial(15))
## 41438 years, 28 weeks, 5 days, 6 hours, 50 minutes, and 24 seconds.
\(15!\) seconds ago it was 40,000 BC in the height of the Upper Paleolithic. The oldest known cave art in Cave of El Castillo, Spain dates back to this time.
how_long_ago(factorial(16))
## 663016 years, 42 weeks, 4 days, 14 hours, 52 minutes, and 48 seconds.
\(16!\) seconds ago is between 700,000 and 600,000 years ago in the Lower Paleolithic. During this period, early humans first settled northern Europe.
how_long_ago(factorial(17))
## 11271285 years, 46 weeks, 6 days, 9 hours, 18 minutes, and 0 seconds.
\(17!\) seconds ago, humans, chimpanzees, and bonobos had not yet diverged evolutionarily, with the first hominin likely appearing some 3-4 million years later.
how_long_ago(factorial(18))
## 202883146 years, 9 weeks, 4 days, 2 hours, 16 minutes, and 48 seconds.
\(18!\) seconds ago Earth was transitioning from the Triassic to the Jurassic within the Mesozoic Era. Terrestrial species lived on a single supercontinent, Pangaea, though many land- and sea-dwelling organisms disappeared following a mass extinction event which paved the way for the domination of Earth by the dinosaurs.
how_long_ago(factorial(19))
## 3854779777 years, 25 weeks, 4 days, 1 hours, 51 minutes, and 28 seconds.
\(19!\) seconds ago (3.8 billion years ago) the Earth was not yet 1 billion years old. On this very young Earth, life as we know it began. Specifically, the first organisms appeared (cells resembling prokaryotes) and several thousand million years later lived the last universal ancestor to which all current life on Earth can be traced back.
how_long_ago(factorial(20))
## 77095595549 years, 42 weeks, 0 days, 8 hours, 53 minutes, and 20 seconds.
\(20!\) seconds ago is impossible because the Big Bang only happened 13.8 billion years ago. In fact, \(20!\) seconds is roughly 5.5 times the age of our universe!