Q&A: Real-Life Application Involving a Non-Trivial Limit

Cross-posted from here.

I teach an introductory calculus course to engineers. Most books & resources I’ve found do a good job of illustrating limits graphically, numerically, and formulaically (where one can do some clever rearranging to see what the limit is). But I haven’t seen any motivating example that shows how we might encounter limits in a real-life application.

Can you provide an interesting, natural, and simple example of some physical/geometrical etc. system, which has a quantity $f(x)$ that is not defined at a finite point $x=a,$ but we would still find it interesting to ask for $\lim\limits_{x\to a} f(x)?$


First thing that comes to my mind is the limit

$$\begin{align*} \lim\limits_{x \to 0} \dfrac{\sin x}{x} = 1. \end{align*}$$

This limit justifies the small-angle approximation $\sin \theta \approx \theta$ (for $\theta \approx 0$) that is widely leveraged in engineering/physics classes to tame unwieldy equations.

(Often, the existence of a sine term prevents the existence of a closed-form solution, but if you replace that sine term with a linear term, then you can get a closed-form first-order approximation of the solution that is sufficiently accurate for practical purposes.)

A concrete and accessible example would be calculating the period of a pendulum: http://www.physicslab.org/Document.aspx?doctype=3&filename=OscillatoryMotion_PendulumSHM.xml

Note that if you want to avoid the need for derivatives and differential equations, then you can ask the question “can you model a pendulum like a spring” instead of “what is the period of a pendulum”.

  • Can you model a pendulum like a spring?
  • Springs follow Hooke's law: the restoring force is proportional to the displacement. But in a pendulum, the restoring force is $mg \sin \theta.$ So, you can't model it like a spring. Right?
  • ... actually, for small angles, $\sin \theta \approx \theta$ and $\textrm{displacement} \approx L\theta,$ so the restoring force is $\approx mg/L \cdot \textrm{displacement}.$ So, you CAN model it like a spring when the angle is small.