Why does the limit of $\lim_{x \to \infty} \arcsin \left(\frac{x+1}{x}\right)$ not exist?

Because $\frac{x+1}{x}>1$ for $x>0$, and $\arcsin{y}$ is not defined for $y>1$.

On the other hand, the limit as $x \to -\infty$ does exist, since $-1<\frac{x+1}{x}<1$ for sufficiently large negative $x$, and is $\pi/2$.

The $\arcsin$ function is only defined on the domain $-1 \le x \le 1$. Since the input ${x+1 \over x} > 1 \,\forall x > 0$, the limit does not exist.

The limit does not exist, because $\frac{x+1}{x}$ approaches $1$ from the right, where $\arcsin(x)$ is not defined.