Question

f'(x)=lim h->0 f(x+h)-f(x)/h, if f(x)= { 2, -10 9 years ago

Answer 1

Is this multiple choice?

Answer 2

Yes it is, I will provide the choices. Give me a second

Answer 3

Okay thanks

Answer 4

Will a PDF work?

Answer 5

With the derivative defined as the limit
\[f'(x):=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]or equivalently (by setting \(h=x-0\),
\[f'(0):=\lim_{x\to0}\frac{f(x)-f(0)}{x-0}\]in order for the function to be differentiable at \(x=0\), this limit must exist.

Given that
\[f(x)=\begin{cases}2&\text{if }-10<x<0\\[1ex]1&\text{if }0\le x<10\end{cases}\]the derivative exists at \(x=0\) if
\[\lim_{x\to0^-}f'(x)=\lim_{x\to0^-}f'(x)\]Is this the case?