prove that $$\dfrac{|x-z|}{|x-y|}=1+\dfrac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$$ for $|x|\longrightarrow \infty$
where $$\hat{x}=\dfrac{x}{|x|}$$ This problem from book,following is my idea:
$$\dfrac{|x-z|}{|x-y|}=\dfrac{|x-y+y-z|}{|x-y|}=\dfrac{\sqrt{|x-y|^2+2(x-y)\cdot (y-z)+|y-z|^2}}{|x-y|}=\sqrt{1+2\dfrac{x-y}{|x-y|}\cdot(y-z)+\dfrac{|y-z|^2}{|x-y|^2}}$$ and use Taylor $$(1+x)^a=1+ax+o(x^2),x\longrightarrow 0$$
then $$\dfrac{|x-z|}{|x-y|}=1+\dfrac{x-y}{|x-y|}\cdot(y-z)+O\left(\left(\dfrac{x-y}{|x-y|}\cdot(y-z)\right)^2\right)$$
But I can't prove $$\dfrac{|x-z|}{|x-y|}=1+\dfrac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$$ for $|x|\longrightarrow \infty$
this problem from
Thank you everyone.