I failed to understand how what do author of my vector calculus textbook arrived at below equation:
Based on equation below: $\lim_{x\to a}\frac{\|f(x)-[f(a)+Df(a)(x-a)]\|}{\|x-a\|}=0$
Given $\|f(x)-f(a)-Df(a)(x-a)\|$
Thus since f is differentiable at a, we can make $\|f(x)-f(a)-Df(a)(x-a)\|$
as small as we can wish by keeping $\|x-a\|$ appropriately small.
In particular,$\|f(x)-f(a)-Df(a)(x-a)\|\leq\|x-a\|$ if $\|x-a\|$ is appropriately small.
Note: I modify above proof from my textbook to fit the context of this question.
Note that the 'D' above is differential symbol.
to be more precise with my question, I was thinking straightforward
In first equation 0 multiply with $\|x-a\|$ should be 0
I'm not sure how the author arrive with $\leq\|x-a\|$ at the last equation.