It is well-known that trigonometric functions oscillate on the real axis and the limit does not exist as the argument approaches infinity.
However, I suspect that a limiting value exist if the argument approaches any complex infinity that is not real, i.e. $$\lim_{r\to\infty} f(re^{i\theta})$$ is suspected to exist for $\theta \ne n\pi$, where $f$ is a trigonometric function.
I confirmed it is the case for $\tan (z)$ by decomposing it into real and imaginary parts.
For real part: $$\lim_{r\to\infty}\frac{\sin 2r\cos\theta}{\cos 2r\cos\theta+\cosh 2r\sin \theta}=0$$ which is straightforward.
For imaginary part: $$\lim_{r\to\infty}\frac{\sinh 2r\sin\theta}{\cos 2r\cos\theta+\cosh 2r\sin \theta}=\text{sgn}(\sin\theta)$$
Thus $$\lim_{r\to\infty}\tan(re^{i\theta})= \text{sgn}(\sin\theta)i$$
My questions are:
Are the above calculations correct?
Is there an easier way to compute the limit, other than decomposing it into real and imaginary parts?
Indeed, the limits should be well known. Are there some reliable references that summarize the results for various trigonometric functions?
Thanks in advance.
You can base all calculations on the fact that $e^{iz} \to \infty$ as $z \to \infty$ with $-\pi+\varepsilon<\arg{z}<\varepsilon$ and $e^{iz} \to 0$ as $z \to \infty$ with $\varepsilon<\arg{z}<\pi-\varepsilon$ (this follows because $\lvert e^{i(x+iy)} \rvert = e^{-y}$). For example, $$ \tan{z} = \frac{e^{iz}-e^{-iz}}{i(e^{iz}+e^{-iz})} = -i\frac{e^{2iz}-1}{e^{2iz}+1}, $$ which tends to $i$ for $\varepsilon<\arg{z}<\pi-\varepsilon$ and $-i$ for $-\pi+\varepsilon<\arg{z}<\varepsilon$.
Similarly, for the other functions, $\cos{z}$ and $\sin{z}$ both tend to $\infty$ for $z \to \infty$ with $\arg{z}$ bounded away from $0$ and $\pi$. Therefore their reciprocals $\csc{z}$ and $\sec{z}$ both tend to $0$ in the same region.
Lastly, $\cot{z}=1/\tan{z}$, so it tends to $-i$ for $\varepsilon<\arg{z}<\pi-\varepsilon$ and $i$ for $-\pi+\varepsilon<\arg{z}<\varepsilon$.