Why Slopes of Perpendicular Lines Are Negative Reciprocals

It was years since I "derived" the result that slopes of perpendicular lines were negative reciprocals of each other. So I thought it would be easy to show that when , where, in Figure 1, is the slope of line (black) and is the slope of line (red). Clearly, lines and are perpendicular when . 

Indeed, Maple immediately calculates 

=  

 and with more care, 

=  

Now how could this be made more transparent for a calculus student? It would certainly help to recall the addition formula for the tangent of the sum of two angles, namely, 

 Heuristically, as , becomes unbounded.  Hence, the limit should easily come out to be , either by the algebraic device of dividing numerator and denominator by or by the calculus, using l'Hopital's rule.   

I decided to try the calculation with the Limit Methods tutor where I had access to a button that would apply l'Hopital's rule. To expedite this calculation, see the steps in Table 1. 

Tools≻Load Package: Student Calculus 1	Loading Student:-Calculus1
Type and press the Enter key.   Context Menu: Expand   Context Menu: Tutors≻Limit Methods≻
Table 1   Launching the Limit Methods tutor

Figure 2 shows the Limit Methods tutor after the limiting value of has been set to . 

Clicking the l'Hopital's Rule button brings up the dialog shown in Figure 3. 

The arrows at the top of the panel shown in Figure 3 allow one to determine how to rewrite a product such as so that l'Hopital's rule would apply. Here, it is only necessary to click the Apply button to receive the message shown in the message window captured in Figure 4. 

Maple claims l'Hopital's rule does not apply because it does not see the input fraction as an indeterminate form at . To determine this, Maple takes the limits of the numerator and denominator because at both are undefined. With no additional information about either or the direction of approach to for , Maple cannot resolve these limits and declares that l'Hopital's rule does not apply. 

Incidentally, it doesn't help to take a one-sided limit by setting the Direction field in the tutor. Indeed, without any assumptions on , the best Maple can do is seen in the two limits in Table 2. 

On the basis of the computed limit of the denominator, Maple concludes that the given fraction is not an indeterminate form, and that therefore l'Hopital's rule cannot be applied. It might be unfortunate that Maple declares l'Hopital's rule does not apply when in reality it's just that Maple can't ascertain that it does. But on the other hand, it is significant that Maple first checks that to see if the rule is applicable before blindly applying it just because the user requested it.