The Unit Step Function and The Dirac Delta

[trfrm]

Wa'alaikumsalam warahmatullahi wabarakatuh.

Excuse me ... .

Can the unit step function,

$$u(x):=\left\{\begin{array}{ll}1&\textrm{if}~x>0\\0&\textrm{if}~x<0\end{array}\right. ,$$

be approached by the function

$$v(x):=\displaystyle\lim_{k\rightarrow\infty}\frac{1}{2}\left(1+\tanh{kx}\right)$$

and

$$w(x):=\displaystyle\lim_{k\rightarrow\infty}\frac{1}{2}\left(1+\frac{2}{\pi}\arctan{kx}\right)$$ ... ?

Next ... , can Dirac Delta $\delta(x)$ be approached by $v'(x)$ and $w'(x)$, such that

$$v'(x):=\frac{dv(x)}{dx}$$ and $$w'(x):=\frac{dw(x)}{dx}$$ ... ?

Thank you very much ... . God bless you all ... . 

Benedictus qui venit in nomine Domini.

[trfrm]

Namo amitabha.

@ river_rat

I have corrected my post ... . I have made a grammatical mistakes in my post, so my post cannot be understanded ... . I'm sorry very much ... .

Wal bi Taufiq wal Hidayah.