Finding limit of norm of difference quotient

Question

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be continuously differentiable and suppose $F(1,1)=(1,0)$. Suppose you are given with the Jacobian matrix of $F$. How can we compute for $\displaystyle\lim_{(x,y)\to(1,1)}\dfrac{\|F(x,y)-F(1,1)\|}{\|(x,y)-(1,1)\|}$? Thanks

Answer 1

From the differziability of $F$ in $(1,1) $ you have that

$\lim_{(x,y)\to(1,1)}\frac{F(x,y)-F(1,1)-J(F)(1,1)((x,y)-(1,1))}{||(x,y)-(1,1)||}=0$

and so you have that

$0\leq\frac{||F(x,y)-F(1,1)||}{||(x,y)-(1,1)||}\leq $

$\frac{||F(x,y)-F(1,1)-J(F)(1,1)((x,y)-(1,1))||}{||(x,y)-(1,1)||} + \frac{||J(F)(1,1)((x,y)-(1,1))||}{||(x,y)-(1,1)||}$

and, if that limit exists, than

$\lim_{(x,y)\to(1,1)}\frac{||F(x,y)-F(1,1)||}{||(x,y)-(1,1)||}\leq ||J(F)(1,1)||$