For simplicity, suppose $X$ is a smooth $n$-dimensional variety defined over $\mathbb{Q}$. Then the etale cohomology of $X$, denoted by $H^i_{\text{et}}(X,\mathbb{Q}_\ell)$, gives a representation of the Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, which is only ramified at finitely many prime numbers. Suppose $p\neq \ell$ is an unramified prime, then there is a well-defined (geometric) Frobenius action on $H^i_{\text{et}}(X,\mathbb{Q}_\ell)$. \begin{equation} \text{Fr}_p:H^i_{\text{et}}(X,\mathbb{Q}_\ell) \rightarrow H^i_{\text{et}}(X,\mathbb{Q}_\ell). \end{equation} Is there a reasonable way to define the limit $$\lim_{p \rightarrow \infty}\text{Fr}_p,$$ which acts on a cohomology space?

The thing that comes to my mind is that the completion of $\mathbb{Q}$ at the infinite prime is $\mathbb{R}$, while the Galois group $\text{Gal}(\mathbb{C}/\mathbb{R})$ is generated by the conjugation $c$, and it has an action \begin{equation} c:H^i(X(\mathbb{C}),\mathbb{R}) \rightarrow H^i(X(\mathbb{C}),\mathbb{R}). \end{equation} I don't know whether it is reasonable to say $$\lim_{p \rightarrow \infty}\text{Fr}_p=c?$$ One thing that confuses me is that the Frobenius contains lots of arithmetic information of $X$, while $c$ does not. Another thing that confuse me is that, when define the full $L$-function of $X$, at the (unramified) prime number $p$, the local $L$-factor is defined by the characteristic polynomial of the Frobenius $\text{Fr}_p$. While at the infinite prime of $\mathbb{Q}$, the local $L$-factor is given by the Gamma function, which (to me) does not look like the "characteristic polynomial" of the complex conjugation. (Even though complex conjugation plays an important part in the definition of the $L$-factor at infinite prime.)