1. prove a limit

Suppose that ;

1)$\displaystyle f: R\rightarrow R^2$ such that : $\displaystyle f(x) =(sinx,cosx)$

2)The Euclidian norm of a vector $\displaystyle v=(u_{1}, u_{2})$ is defined as :

$\displaystyle ||v||_{Eu} =\sqrt{ u_{1}^2 +u_{2}^2}$

3) The maxnorm of a vector $\displaystyle v=(u_{1},u_{2})$ is defined as :

$\displaystyle ||v||_{max} = max( |u_{1}|,|u_{2}|)$

Where $\displaystyle u_{1},u_{2}$ belong to the real Nos R

Then prove :

$\displaystyle \lim_{x\to 0} f(x) = (0,1)$ ,with respect to both norms

I know that i have to prove the following:

1) given ε>0 ,i got to find a δ>0 ,such that :

if 0<|x|<δ , then $\displaystyle ||(sinx,cosx)-(0,1)||_{Eu}<\epsilon$

2)given ε>0 ,i must find a δ>0 ,such that:

if 0<|x|<δ ,then $\displaystyle ||(sinx,cosx)-(0,1)||_{max}<\epsilon$ .

Any suggestions for that delta??

2. Originally Posted by alexandros
Suppose that ;

1)$\displaystyle f: R\rightarrow R^2$ such that : $\displaystyle f(x) =(sinx,cosx)$

2)The Euclidian norm of a vector $\displaystyle v=(u_{1}, u_{2})$ is defined as :

$\displaystyle ||v||_{Eu} =\sqrt{ u_{1}^2 +u_{2}^2}$

3) The maxnorm of a vector $\displaystyle v=(u_{1},u_{2})$ is defined as :

$\displaystyle ||v||_{max} = max( |u_{1}|,|u_{2}|)$

Where $\displaystyle u_{1},u_{2}$ belong to the real Nos R

Then prove :

$\displaystyle \lim_{x\to 0} f(x) = (0,1)$ ,with respect to both norms

I know that i have to prove the following:

1) given ε>0 ,i got to find a δ>0 ,such that :

if 0<|x|<δ , then $\displaystyle ||(sinx,cosx)-(0,1)||_{Eu}<\epsilon$

2)given ε>0 ,i must find a δ>0 ,such that:

if 0<|x|<δ ,then $\displaystyle ||(sinx,cosx)-(0,1)||_{max}<\epsilon$ .

Any suggestions for that delta??
Try looking at when $\displaystyle \sin(x),1-\cos(x)$ intersect. You obviously want to use $\displaystyle \arcsin(\varepsilon)$. But, what must you impose first?

3. I am sorry i do not understand,but if you suggest a delta i can try to prove the problem

Then you can explain how you got that delta

4. Can anyone help me ,please??

5. perhaps the problem is not derivable??