@Jomo
I am not being argumentative. I am genuinely curious. How do you define the difference quotient? Does it not require the denominator to be non-zero just as all division permissible in the real numbers requires a non-zero denominator?
Yes, it does. The difference quotient, \(\displaystyle \frac{f(x+h)- f(x)}{h}\) requires that \(\displaystyle h\ne 0\). However, the derivative is NOT the difference quotient, it is the
limit of the difference quotient as h goes to 0.
I once made the mistake of agreeing to teach a course titled "Calculus for Economics and Business Administration" in the Business Administration Department. The only prerequisite for the course was basic algebra and I had to use the textbook assigned by the Business Admimistration Department. It was awful! One one page they gave the basic rules for limits:
"The limit of f+ g is the limit of f plus the limit of g."
"The limit of f- g is the limit of f minus the limit of g."
"The limit of fg is the limit of f times the limit of g."
"The limit of f/g is the limit of f divided by the limit .of g
provided the limit of g is not 0."
The very next page defines the derivative as "\(\displaystyle \lim_{h\to 0}\frac{f(x+h)- f(x)}{h}\)" completely ignoring the fact the rule for f/g does not apply!
(If you are wondering, we need one more rule: if for some interval (a, b) that includes \(\displaystyle x_0\) f(x)= g(x) for all x
except \(\displaystyle x_0\) (a "punctured interval") then \(\displaystyle \lim_{x\to x_0} f(x)= \lim_{x\to x_0) g(x)\).)