I want to address the drivel in Folland's article at:
https://sites.math.washington.edu/~folland/Math134/lin-approx.pdf
He writes [f(a+h)-f(a)]/h = f'(a) + E(h) and ignorantly calls E(h) the error term without even defining it.
f'(a) + Q(a,h) are the components of the SLOPE as outlined in my historic theorem:
https://drive.google.com/open?id=1RDulODvgncItTe7qNI1d8KTN5bl0aTXj
Thus, there is NO error term whatsoever. The sum of the components f'(a) and Q(a,h) is ALWAYS the SLOPE of the NON-PARALLEL secant line in your bogus calculus. Both components are well defined by lengths of triangles in my historic geometric theorem.
Why the obsession with "error terms"? Well, long before my closed form Gabriel Polynomial, there was the inferior Taylor polynomial which ALWAYS is defined with an error term. Then there were numerous other flavours of formula with error terms. Mainstream academics got used to and started loving error terms in their ignorance, arrogance and stupidity. So the syphilitic thought of "error term" plus "limit theory" produced the perfect storm that resulted in your bogus mainstream formulation of calculus.
Now back to Folland's drivel...
The second component Q(x,h) is a function of both x and h, not just h alone.
Next, Folland drivels that E(a,h) tends to 0. Really? E(a,h) is a CONSTANT for any a and any h. It DOES NOT change or tend to anything.
And finally he arrives at the drivel of SPEED which has nothing to do with the topic at all. He drivels that E(a,h) changes faster than h as both "head" to 0.
Since he doesn't define speed, I assume he means the average for each of h and E(a,h) for any given fixed intervals of x. This syphilitic thought has found its way in all of calculus and was propagated by that fucking moron Prof. Gilbert Strang (MIT).
Now consider that for the function f(x)=x^2, such a "speed" is the same because h=h and E(a,h)=h. Can one get any dumber? What the fuck does speed have to do at all with any function? NOTHING is changing in the function. One calculating finite differences on fixed intervals doesn't mean shit.
Folland concludes his drivel by stating that his function T is a good linear approximation to the function f because the (imagined) error tends to 0 faster than h tends to 0. In other words, the "approximation" to f(x)=x^2 can't be a good one because h tends to 0 as fast as h tends to 0. LMAO!!!!!!!!!!
Mainstream academics are incorrigibly stupid fucks. I mean this is the shit I have had to deal with in my reeducation of the mainstream math orangutans who NEVER understood the concept of number, much less the intricacies of calculus.
Stupid, stupid, very stupid Franz Fritsche and Jean Pierre Messager and Jan Burse and all their fellow apes. Tsk, tsk.
Still think your bogus calculus is rigorous? LMAO.
The Church of Academia absolutely hate my recent theorem (January 2020) which was realised as a result of my New Calculus now over 30 years old.
Among other things, it proves the mainstream formulation to be an utter fraud.
It also proves that all of calculus can be done just fine without limit theory and bullshit concepts of infinity and infinitesimals.
https://drive.google.com/open?id=1uIBgJ1ObroIbkt0V2YFQEpPdd8l-xK6y
Mainstream orangutans are deeply embarrassed and don't know where to turn. For years the fucking morons have been libeling me and calling me all sorts of names, but the ultimate victory is mine and he who laughs last always laughs best. :-))
There is much more embarrassing news for them in the future ... all in due time, in due time... Chuckle.