it's seems that $\displaystyle\lim _{s\to 1}\sin²(\zeta(s))+\cos²(\zeta(s)) $ dosn't exist as shown here by wolfram alpha , and because $\zeta(1)$ is undefined , My question is to seek if the titled equation has any geometric interpretation .
Question: What is the geometric interpretation of this : $\displaystyle\lim _{s\to 1}\sin²(\zeta(s))+\cos²(\zeta(s)) $ ?
Note:$\zeta$ is the Riemann zeta function
Notice that for $s \ne 1$, it is trivially true that $\cos^2 \zeta(s) + \sin^2 \zeta(s) = 1$ simply becaused $\sin^2 + \cos^2 = 1$. Since $\zeta$ is not defined in $s = 1$, then $\cos^2 \zeta(s) + \sin^2 \zeta(s) = 1$ is not defined in $s=1$ either. It follows that $\lim _{s \to 1} \cos^2 \zeta(s) + \sin^2 \zeta(s) = \lim _{s \to 1} 1 = 1$, while $\cos^2 \zeta(1) + \sin^2 \zeta(1)$ simply does not exist.
The situation is like the following: consider $f : \Bbb R \setminus \{0\} \to \Bbb R$ given by $f(x) = \frac 1 x$, and $g : \Bbb R \to \Bbb R$ given by $g(x) = 1$. One has that $\lim _{x \to 0} (g \circ f) (x) = \lim _{x \to 0} 1 = 1$, but $(g \circ f) (1)$ does not exist.