It all started when I tried to get an interesting variation on graphs of inverse trigonometric functions. Tiring of constant scale changes and translations of inverse trig graphs, I tried , thinking that this product of odd functions leading to an even function would be a nice, but minor, extension for my students.
I reasoned that because the magnitude of arctangent approached as , the graph of must approach . As shown and to my surprise, seemed to parallel the anticipated absolute value function instead of approaching it. Hmmmm…..
If this is actually true, then the gap between and must be constant. I suspected that this was probably beyond the abilities of my precalculus students, but with my CAS in hand, I (and they) could compute that limit anyway.
Now that was just too pretty to leave alone. Because the values of x are positive for the limit, this becomes .
So, four things my students should see here (with guidance, if necessary) are
- actually approaches ,
- the limit can be expressed as a product,
- each of the terms in the product describes what is happening to the individual terms of the factors of as x approaches infinity, and
- (disturbingly) this limit seems to approach . A less-obvious recognition is that as , must behave exactly like because its product with x becomes 1,
But what do I do with this for my precalculus students?
NOTE: As a calculus teacher, I immediately recognized the product as a precursor to L’Hopital’s rule.
and this form permits L’Hopital’s rule
OK, that proves what the graph suggests and the CAS computes. Rather than leaving students frustrated with a point in a problem that they couldn’t get past (determining the gap between the suspected and actual limits), the CAS kept the problem within reach. Satisfying enough for some, I suspect, but I’d love suggestions on how to make this particular limit more attainable for students without invoking calculus. Ideas, anyone?