Let $\Delta x = x_1 – x$, where $x$, $x_1$ are real numbers.
Let $f$ be a real valued function; we define the difference quotient as: $$\frac{\Delta f}{\Delta x} = \frac{f(x + \Delta x) – f(x)}{\Delta x} \text{.}$$
I've read this in a physics book:
When $\Delta x \to 0$, the denominator of the difference quotient tends to zero; in order for the limit $\lim_{\Delta x \to 0}{\frac{\Delta f}{\Delta x}}$ to exists (and be finite), also the numerator must tend to zero. That is, $\lim_{\Delta x \to 0}{f(x + \Delta x) = f(x)}.$
Is the book saying that the limit of a quotient is finite when the denominator tends to zero iff the numerator tends to zero too?
If so, why this happens?
Best Answer
To answer your question: No. The book doesn't try to say "iff". The book says that the limit of a quotient is finite when the denominator tends to zero only if the numerator tends to zero too. (Note I replaced "iff" with "only if" in your sentence.) Only one direction follows directly, the other doesn't, and for some functions and some choice of $x$ it is true, for others it is false.