Is there somone who can show me what $\; \lim_{x\to\infty} (\arccos x) =i\infty \;$ means?
Does it meant that limit does not exist? $\:$ If yes, how can one prove that limit does not exist?
Note : $\arccos$ is the inverse function of $\cos$
Thank you for any kind of help
The real function $\arccos x$ is defined for $-1 \le x \le 1$. Not for $x>1$. But the complex function $\arccos x$ is defined for all $x$. If $w = \arccos x$, then $\cos w = x$, so that $$ \frac{e^{iw}+e^{-iw}}{2} = x $$ and we may solve a quadratic equation to see $$ \arccos x = i \log\left(x+\sqrt{x^2-1}\;\right) $$ when $x>1$. Can you do the limit of this as $x \to \infty$?
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As $x \to +\infty$ along the real axis, $\arccos x$ goes to $i\infty$, which we can see by watching it follow the imaginary axis upward in the complex plane.