How does the idea of limits make the calculus precise?
Imagine driving on the Bonnet Carre Spillway Bridge in Louisiana. This 10-mile-long structure is often
sparsely populated deep in the night. On one such night you daringly decide to accelerate from your current
55 mph, through the speed limit of 60 mph, all the way to 65 mph, in time T seconds, from where you
maintain the constant speed as you drive off the bridge.
We are interested in the mathematics – more specifically the calculus – of this situation. Much like how
geometry is a study of shapes and angles, algebra involves arithmetic operations and calculations, calculus is
the study of continuous changes. We know that the acceleration described above is a continuous change.
Evidently, vehicle’s speedometer would have smoothly gone from 55 to 60 without any pauses or breaks. As
speed is continuous, not discrete which suggests that the cars’ speed has to have a whole number value, we
can say that the during time T the car’s speed held every numerical value from 55 to 65 mph, even if it was
instantaneously. This, for example, can include 19.854748 mph. Therefore, at one instantaneous moment in
time it must have been 60 mph – the moment you crossed the speed limit.
Yet, how can the speed be a stationary value for a moment in time if the car is constantly accelerating.
We can think about this situation in terms of graphs. Figure 1 is a distance-time graph. The velocity of a
travelling object can be calculated from its distance-time graph. This is done by simply dividing the distance
covered by the time taken to cover that distance; however, this is only accurate when the velocity is
constant. When the velocity is changing, like figure 1, so will the gradient signifying that same distance is
covered in shorter and shorter time periods. In such a case, how can we
calculate the gradient?
The gradient of the line is constantly changing, as the line is curved.
Similarly, due to the car’s acceleration, the velocity is continually changing.
How can we pinpoint the exact point in the graph where the gradient is 60
mph when it is continually changing?
It is these kinds of problems that form the foundation, and led to the fruition,
of calculus.
It was Newton and Leibniz who attempted to solve this problem, and
consequently formulated calculus.
Figure 1: Example of a distance – time
graph [1]
Both of them used the idea of ‘infinitesimals’ to develop an answer to these problems. Infinitesimals is a
non-zero quantity that is smaller than any real number. It was crucial that an infinitesimal be non-zero as
you can never divide by zero [1]. Division by zero is undefined simply because of the following:
𝑛𝑥0=0
Where n is any number. If dividing by zero gave us a numerical answer, it would contradict the universally
accepted statement above:
𝑎
=𝑛→𝑛𝑥0=𝑎
0
The use of infinitesimal is simple and founded on basic knowledge of mathematics. Speed is worked out by
dividing the distance travelled by time (taken to travel that distance) – gradient of a distance – time graph. In
a situation where the speed is continually changing, we can take a shorter time period (around the time that
the speed went past 60 mph) and divide it by the distance travelled, which will be smaller. However, it can
still be argued that as the speed is changing, the answer obtained is inaccurate. The only way to achieve
greater and greater accuracy is by decreasing the time. Why? Because the ‘problem disappears’ if we take
shorter time increments – short enough so that the speed has no time to change [2]. As the speed has no time
to change, it must not be accelerating during that miniscule period of time, allowing us to obtain the precise
speed at that time. For this we would have to measure the distance car travelled in an ‘infinitely small period
of time’ – an infinitesimal [2]. This time cannot be zero as this division is
undefined as previously established.
For the graph above, the gradient is calculated by the change in distance,
∆d, divided by change in time, ∆t. In order to find the exact gradient at a
point, where the speed is 60 mph, we would have to use smaller and
smaller – infinitesimal – changes in distance and time so that the curve
has no space to change direction. This process is illustrated in Figure 2:
The main principle behind these calculations is the fact that the
acceleration is not allowed to change too much in an incredibly short time period dubbed an ‘infinitesimal’.
Figure 2: Differentiation from first principles [3]
Newton and Leibniz worked on the same problem, although separately,
coming to the same conclusion. Calculus is divided in to two branches: integration and differentiation.
Integral calculus calculates the area bounded by a curve and the x-axis for any function, it follows the same
principles of using infinitesimals, but we will not be focusing on integration for this problem.
Our problem is closely related to differential calculus which allows us to calculate rates of change. It can
calculate the slope of a curve at any point using the derivative of a function. A function is a mathematical
equation that numerically describes behaviours. The function of speed as it varies with distance and time is
𝑑
the following: 𝑠(𝑑, 𝑡) = where d and t act as function variables. So, for any function, infinitesimal calculus
𝑡
allows us to find the slope at any point, even if it is curved.
Obviously, infinitesimals do not really exist as we do not know of any number that is smaller than all real
numbers while being non-zero. But Newton and Leibniz found great success using this convenient concept.
Calculus gave amazingly accurate solutions that no one thought to object it. Of course, calculus’s accuracy
speaks volumes about its validity; nonetheless, the idea of infinitesimal incited criticism among academics.
One very vocal critic was Lord Bishop Berkeley, an Irish philosopher. He published his disapproval in his
book The Analyst where he said:
‘And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent
increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not
call them ghosts of departed quantities’
He argued the ‘logical fallacy’ of calculus and questioned the rationality of these ‘fluxions’ – these
instantaneous rates of change; which are calculated through ‘vanishing increments’ – infinitesimal [3]. His
arguments stated that an infinitesimal simply cannot exist, and that its definition changes based on
convenience. An infinitesimal has to be vanishingly small, yet finite. No vanishingly small number can be
fixed, otherwise there will definitely be a smaller number. No matter how incredibly small a number is, if
you halve it or even take away one, you will get an even smaller value due to the fundamental properties of
numbers and how they work.
His phrase: ‘ghosts of departed quantities’ became famous among the community and was a logical analysis
of Newton and Leibniz’s infinitesimal. It raised important questions against the existence of infinitesimals –
the comparison highlighted the unstable reality of this vanishingly small quantity. The reality is that this
quantity has to satisfy many conditions, some of which are contradictory. An infinitesimal must be ‘smaller
than any imaginable positive number’, must not be zero, while also remaining fixed and real [6].
Berkeley’s arguments were grounded and crucial in pointing mathematicians toward a more precise
calculus.
It was many decades later that calculus was made rigorous and the concept was clarified. It was the work of
Cauchy, Weierstrass, and Riemann that brought calculus a ‘logical footing’ [4]. The conclusion was simple:
define calculus in terms of limits rather than infinitesimals.
Instead of needing vanishingly small increments, we simply need to find a limit that our solutions get closer
and closer to as we use small and smaller values. This is analogous to the definition of the square root of 2.
Like any number, the number two has a square root.
𝑥 2 = 2 where 𝑥 = √2
However, x is a limit of a sequence of finite digits – an infinite decimal. It is nearly impossible to compute
or calculate all the digits of x but we know that it squares to two. Therefore, the more digits we take of x, the
closer we will be to two. We will never get an answer that is exactly two, but our answers will keep getting
closer. Hence, two is the limit.
Weierstrass, a key figure in the reform of calculus, used a similar argument to define limits and make
calculus precise [5]. If we take the following sequence of numbers:
1 1 1 1
1, , , , , …
2 3 4 5
The sequence is arguably infinitely large; however, infinity is now defined as a limiting process. As the
sequence tends to infinity, the sequence ‘tends to the limit 0’ [5]. Using the same process for calculating the
𝛿𝑑
gradient on a graph, as the change in distance and time becomes infinitely small, the answer, should tend
𝛿𝑡
to a limit which is a close approximation for the speed (60 mph). This calculation gives us a good
approximation for when exactly and at what distance the speed is approximately 20 mph.
If, for the sequence above, you want a number that is 0.0000001 close to zero, you simply make the
denominator bigger than ten million. By continually increasing the denominator we keep getting closer and
closer to zero while never reaching zero. Therefore, zero must be the limit and by deciding upon how close
we want our sequence to be to zero, we can get an answer that follows the accuracy required.
To better understand this, it is important to emphasise that infinity is no longer a number but rather a
limiting process. In calculus, if something is infinite, it has no limit. For example, for the function 𝑓(𝑥) =
1
, as x becomes infinitely small and approaches zero, the limit is infinity – there is no value or limit. For
𝑥
1
another function 𝑓(𝑥) = 1−𝑥 the limit as x becomes infinitely small is 1. So, infinity stands for a process
through which we can get a close approximation but never the exact answer.
Instead of relying on the unstable and debatable infinitesimal, we have a just way of defining calculus that
stands all tests. Instead of trying to exactly calculate a derivative using infinitesimals, we settle for a good
estimate – an estimate that can be improved if we, for example, take more digits (square root of two).
Limits are used in the renowned formula that is the face of differential calculus. The formula that allows us
to calculate gradients of tangents and helps determine instantaneous rates of change.
lim (𝑥 + 𝛿𝑥) − 𝑥
𝛿𝑦 𝛿𝑥→0
𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = lim
=
𝛿𝑥→0 𝛿𝑥
(𝑦 + 𝛿𝑦) − 𝑦
The key in this formula is limits. 𝛿𝑥 is never explicitly said to equal zero, rather the derivative is the limit as
𝛿𝑥 approaches zero. Whereas before we would have used the idea that 𝛿𝑥 equals the infinitesimal – an
inaccurate idea as explained.
Going back to you on a serene night in Louisiana, the exact moment you break the speed limit along with
the exact distance travelled at this moment cannot be calculated but using differential calculus we can get a
close estimate.
The principle idea is the same. We use a very short time interval, 𝛿𝑡 – so short that the car does not have
time to accelerate much. The car would have travelled a very small distance during 𝛿𝑡. This 𝛿𝑡 is not zero
nor an infinitesimal, rather we use the idea that the limit as 𝛿𝑡 approaches zero gives us the instantaneous
speed.
𝛿𝑑
𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑠𝑝𝑒𝑒𝑑 =
𝛿𝑡
lim (𝑡 + 𝛿𝑡) − 𝑡
𝛿𝑑 𝛿𝑡→0
𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑠𝑝𝑒𝑒𝑑 = lim
=
𝛿𝑥→0 𝛿𝑡
(𝑑 + 𝛿𝑑) − 𝑑
We no longer need to ‘worry about the infinite or the infinitely small’. Using shorter and shorter time
intervals, we can achieve a ‘closer and closer approximation’ [2]. Similar to the square root of two example,
we will never exactly know when the speed is exactly crosses the speed limit, nor what the speed is at a
specific time as it is constantly changing but using smaller intervals, we can get close to the answer. The
solution will either be slightly above the required value or below, but never exactly equal. But this is not a
matter to worry about.
As Gower concisely explains in his book, an introduction to Mathematics, the precision we require can be
easily achieved. As long as we define the ‘margin of error’, i.e. how close our answer can be to the true
value, by choosing a small enough value of 𝛿𝑡 we can acquire an answer that is within the margin of error –
answer is ‘as close to t as the margin allows’ [2]. The problem is made more precise and elegant by
removing the notion that 𝛿𝑡 needs to be an infinitesimal – instead of dealing with the complexities of
infinity, we are simply approximating our solution. Much like how the square root of two is an infinite
decimal – a limit of a sequence of infinite decimals – instantaneous speed is the limit as 𝛿𝑡 approaches zero.
Newton and Leibniz’s calculus was remarkably effective, and reliable. Their formulae gave the right
answers and proved their legitimacy again and again. However, on a deeper and a more philosophical level,
the foundation of calculus was built on unsteady grounds. When calculus was redefined in terms of limits, it
was not only made more elegant but more precise. Yes, it gives the same answers as it did when the
infinitesimal was in action, but it no longer raises difficult questions as ones that have been highlighted.
Thus, we have more faith in the solutions. Though it might be unnerving to know that we will never have
the exact answer, our extremely close approximations have been just as useful in real-life application of
differential calculus – optimisation problems and rates of change.
References
[1] BBC (2014) Forces for Transport. Last Accessed: 14/12/18
[2] Gowers, T. (28 November 2002) Mathematics: A Very Short Introduction. USA: Oxford University
Press
[3] Revision Maths (no date) Differentiation from First Principles. Last Accessed: 14/12/18
[4] (no author) (no date) The History of Calculus. Last Accessed: 14/12/18
[5] (no author) (10/04/08) Attacks on the Foundation of Calculus. Last Accessed: 14/12/18
[6] Stewart, I. (29/04/95) Tribute to the Infinitesimal. Last Accessed: 14/12/08
[7] Engelhardt, N. (no date) Adam Dickinson: The Ghosts of Departed Quantities Last Accessed: 14/12/18