Algebraic function that acts like $\sin(1/x)$ near zero. (or non-trig function)

Question

Trying to construct an example for a Business Calculus class (meaning trig functions are not necessary for the curriculum). However, I want to touch on the limit problem involved with the $\sin(1/x)$ function.

I am sure there is a simple function, or there isn't... But would love some insight.

I also understand that the functions that satisfy this condition are maybe way outside the scope of the course. I'm just looking for different "flavors" of showing limits that don't exist besides just showing the limit from the left and the limit from the right does not exists.

Answer 1

As you note, this is really outside the scope of a business calculus syllabus. I might argue that anything more than a very informal discussion of limits is too.

In any case I think your business calculus students could profit from understanding that functions need not come from formulas. You can convey lots of the meaning and usefulness of calculus just with sketches of graphs. For this example you could sketch the graph near the origin at large magnification to show the infinitely many oscillations. If you draw the oscillations between the lines $y = \pm x$ you can get continuity. Between $y = \pm x^2$ you get differentiablity too.

Answer 2

Consider applying the fractional part function to $1/x^2$ or something similar. This would be an even function, so the behavior from the left is the same as the behavior from the right of zero, but neither limit from above or below exists because of oscillations.

Note that defining the fractional part function on negative numbers is done differently by various authors, so that's another reason to introduce $1/x^2$ and avoid that ambiguity.

I wouldn't call this an algebraic function, although the notions of integer part and fractional part of positive reals numbers should be pretty intuitive for your "Business Calculus" students. The fractional part function is not continuous, so it is scarcely surprising when limits involving it fail to exist.

Answer 3

I endorse Ethan's answer: define the function by drawing a graph. You know that it's $ x \mapsto \sin ( 1 / x ) $, but they don't need to know that.

If you don't like that, hardmath has given an answer; but you can make it look more like $ \sin ( 1 / x ) $ (including being continuous) by using $ 2 \{ 1 / x \} - 1 $ when $ [ 1 / x ] $ is odd and $ 1 - 2 \{ 1 / x \} $ when $ [ 1 / x ] $ is even. Your students might prefer a more explicit piecewise-defined representation: $$ \cases { 1 - 2 / x & for $ x > 1 $, \\ 2 / x - 3 & for $ 1 / 2 < x \leq 1 $, \\ 5 - 2 / x & for $ 1 / 3 < x \leq 1 / 2 $, \\ \vdots & etc. } $$ This is based on a piecewise-linear approximation of the sine function (or rather a cosine function with period $ 2 $). You can make it differentiable by using a piecewise-quadratic approximation instead, twice differentiable using a piecewise-cubic, etc. (But to make it infinitely differentiable, you're back at the sine function.)