Hello, I recently had a calculus test in Highschool about curve sketching. Our class was allowed cheat sheets so we could write formulas and stuff but I decided to do a "no cheat sheet run" for my last year of highschool cause why not. During the test, I was given the equation f(x) = 4x/x^2+2x+1. But I completely forgot how to do the quotient rule and I didn't want to spent a lot of time trying to use the difference quotient to figure out the shortcut since I had around 15 minutes left and I've to still figure out all the points of the graph. So I just accidentally balled it and did something similar to L'hopital's rule.

When I did this, I got the derivative of 4/2x+2 which then I tested values of b/a of the denominator (b being the possible factors of 2x, and a being possible factors of the constant term, 2x. ) which gave interesting points. Obviously I didn't knew the answer, but given the points alone from the possible values of 1, -1, 2, -2. You can actually find the max and min points of where both the main function and derivative intercept. (But in this case, there was a limit at x = -1 from x^2+2x+1 = (x+1)^2).

I tried with different values of the rational function on desmos (few days after the test) that doesn't meet L'Hopital rules of 0/0 or inf/inf. And you basically always get some points of min and max that either has a point or a limit. You also find some stuff changing the constant term of the antiderivative of the function. Usually getting the derivative function to be some sort of value or limit that is between the antiderivative.

Example of this would just be changing the C value of the function (4x/x^2+2x+1). in this basic function, we know that there's a limit at x = -1. If we change it to something like -100 or -50 on the denominator, you can also find points of inflections. If you change it to something like +5 or +10 of the denominator, you get the min and max points and also x values being the same but both values are either positive or negative on 2 different y values. Something similar as well with changing the C value of the numerator.

I found this as a interesting relationship between L'Hopital's rule and Curve Sketching and was wondering why this happens?