Can tell sign of derivative through observing numerator and denominator limits?

Asked Nov 08 '19 at 00:44

Active Nov 08 '19 at 00:52

Viewed 23 times

Say a derivative is differentiable in the interval $x \in [a,b]$

The derivative of the limit at $f(b)$ is $\lim\limits_{x \to b^+}\frac{f(x)-f(b)}{x-b}$

If we can show that $\lim\limits_{x \to b^+}f(x)-f(b)$ and $\lim\limits_{x \to b^+}x-b$ both converges to $0$ from above, then is $f'(b)\geq 0$?

edited Nov 08 '19 at 00:52

asked Nov 08 '19 at 00:44

Bill Cui

1

Do you think that $x-b$ can converge to $0$ from above when $x\to b$? – bjorn93 Nov 08 '19 at 00:48
Sorry typo! Fixed it – Bill Cui Nov 08 '19 at 00:52
How do you define "a function converges to $0$ from above"? I don't think I've seen such a concept before. – bjorn93 Nov 08 '19 at 01:10
The function $y = 1/x$ converges to $0$ from above as x approaches positive infinity – Bill Cui Nov 08 '19 at 01:17

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