I'm not very experienced with double limits and now I have to evaluate the limits $$ \lim_{x \to \infty,\ y \to \infty} \left[ \tanh(x+2y) + \tanh(x-2y) \right] $$ $$ \lim_{x \to \infty,\ y \to \infty} \left[ \tanh(x+2y) - \tanh(x-2y) \right] $$ and I'm really not sure how to proceed with this. Naturally the first terms are $$ \lim_{x \to \infty,\ y \to \infty} \tanh(x+2y) = 1 $$ but that's as far as I have gotten.
Does this limit exist?
EDIT: So after a little work I can show that $$ \tanh(x-2y) = \frac{\sinh(x-2y)}{\cosh(x-2y)} = \frac{e^{x-2y} - e^{-(x-2y)}}{e^{x-2y} + e^{-(x-2y)}} = \frac{e^{-2y} - e^{-2x}e^{2y}}{e^{-2y} + e^{-2x} e^{2y}} $$ which for constant $y < \infty$ behaves as $$ \lim_{x \to \infty} \tanh(x-2y) = \frac{e^{-2y} - e^{-2x}e^{2y}}{e^{-2y} + e^{-2x} e^{2y}} = \frac{e^{-2y} - 0 \cdot e^{2y}}{e^{-2y} + 0 \cdot e^{-y}} = 1 $$ but also $$ \lim_{y \to \infty} \tanh(x-2y) = \frac{e^{x-4y} - e^{-x}}{e^{x-4y} + e^{-x}} = \frac{0 - e^{-x}}{0 + e^{-x}} = - 1 \mathrm{.} $$
However as $y \to \infty$ AND $x \to \infty$ there's the term $e^{-2x}e^{2y}$ where the two terms tend to zero and infinity. What is the specific reason that allows me to say that this limit doesn't exist?
Notice that
$$\begin{align}\tanh(x+2y)+\tanh(x-2y)&=\frac{\sinh(x+2y)}{\cosh(x+2y)}+\frac{\sinh(x-2y)}{\cosh(x-2y)}\\&=\frac{\sinh(x+2y)\cosh(x-2y)+\sinh(x-2y)\cosh(x+2y)}{\cosh(x+2y)\cosh(x-2y)}\\&=\frac{\sinh(2x)}{\cosh(x+2y)\cosh(x-2y)}\\&=\frac{\frac{e^{2x}-e^{-2x}}2}{\frac{(e^{x+2y}+e^{-(x+2y)})(e^{x-2y}+e^{-(x-2y)})}4}\\&=2\times\frac{1-e^{-6x}}{(e^{2y}+e^{-2x-2y})(e^{-2y}+e^{-2x-2y})}\end{align}$$
Then note that as $x\to\infty$ and $y$ is held constant, then it goes to $2$. Likewise, as $y\to\infty$ and $x$ is held constant, then it goes to $0$. Thus, the limit doesn't exist.