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**HallsofIvy** I once taught a course called "Quantitative Methods for Economics and Business Administration" using a text book assigned by the Business Administration department. On one page of this book we were given three laws of limits:

1) If $\displaystyle \lim_{x\to a} f(x)= L$ and $\displaystyle \lim_{x\to a} g(x)= M$ then $\displaystyle \lim_{x\to a} (f+g)(x)= L+ M$

2) If $\displaystyle \lim_{x\to a} f(x)= L$ and $\displaystyle \lim_{x\to a} g(x)= M$ then $\displaystyle \lim_{x\to a} fg(x)= LM$

3) if $\displaystyle \lim_{x\to a} f(x)= L$ and $\displaystyle \lim_{x\to a} g(x)= M$ **and** M is not 0, then $\displaystyle \lim_{x\to a} \frac{f(x)}{g(x)}= \frac{L}{M}$

On the very next page, they introduced the derivative as the limit of the difference quotient: $\displaystyle f'(a)= \lim_{h\to 0}\frac{f(a+h)- f(a)}{h}$, completely ignoring the fact that the denominator necessarily goes to 0 so this limit could NEVER