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32 votes

"Real life" examples of limits of functions at finite points

First thing that comes to my mind is the limit $$\lim\limits_{x \to 0} \dfrac{\sin x}{x} = 1.$$ This limit justifies the small-angle approximation $\sin \theta \approx \theta$ (for $\theta \approx 0$) ...
Justin Skycak's user avatar
20 votes

"Real life" examples of limits of functions at finite points

"interesting, natural and simple" Illustrating something dynamically might make things interesting. For example, a geometry problem involving a limit (from an old calculus book): Consider a ...
Nick C's user avatar
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4 votes

"Real life" examples of limits of functions at finite points

Here are some examples from a 2015 paper by Steven Jones on students' reasoning about limits. If you pluck a stringed instrument (like a guitar or violin), the pitch of the note ($f$) depends on the ...
Justin Hancock's user avatar
3 votes

What would constitute as a good justification of why a divergent limit is divergence for highschool teaching?

You can rewrite $\displaystyle \lim_{x\to -\infty} \frac{x}{e^x}$ as $\displaystyle \lim_{x\to \infty} -x e^x$ by "reversing the x-axis". i.e. making the $x\to -x$ substitution. Then things ...
Adam's user avatar
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2 votes

"Real life" examples of limits of functions at finite points

Suppose you have an event with tiny but positive probability $p$. As an example, you have $100$ ping-pong balls, all white except for $1$ golden ball, and you choose among them with equal probability, ...
Lee Mosher's user avatar
2 votes

"Real life" examples of limits of functions at finite points

Polynomials are everywhere, and in engineering or physics, it is extremely common to use linear approximations to them $$ \begin{align} \lim_{x \rightarrow 0} \frac{(x + 1)^d - 1}{x} = d \end{align} $$...
Passer By's user avatar
  • 121
1 vote

"Real life" examples of limits of functions at finite points

I'll share an idea, but it requires geometric series, a topic typically covered at the end of a Calculus II course. Still, you might consider using this as a way to preview some future work in ...
Brendan W. Sullivan's user avatar

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