Limit of a vector of two variables

[Dustinsfl]

\begin{align*}
\mathbf{u}_1 &= h_u\mathbf{U}_u\\
&= \frac{a\left(\sinh(u)\cos(v)\unit{i} +
\cosh(u)\sin(v)\unit{j}\right)}{a\sqrt{\cosh^2(u) - \cos^2(v)}}\\
&= \frac{\sinh(u)\cos(v)\unit{i} + \cosh(u)\sin(v)\unit{j}}
{\sqrt{\cosh^2(u) - \cos^2(v)}}\\
\mathbf{u}_2 &= h_v\mathbf{U}_v\\
&= \frac{-a\left(\cosh(u)\sin(v)\unit{i} +
\sinh(u)\cos(v)\unit{j}\right)}{a\sqrt{\cosh^2(u) - \cos^2(v)}}\\
&= \frac{-\cosh(u)\sin(v)\unit{i} + \sinh(u)\cos(v)\unit{j}}
{\sqrt{\cosh^2(u) - \cos^2(v)}}
\end{align*}

I want to take the limit as \(u\to\infty\) and show that \(\mathbf{u}_1\sim\cos(v)\unit{i} + \sin(v)\unit{j}\) and \(\mathbf{u}_2\sim -\sin(v)\unit{i} + \cos(v)\unit{j}\).

I did this but I don't like just neglecting the v terms and then just putting it back in as if nothing happened.

\begin{align*}
\lim\limits_{u\to\infty}\frac{\sinh(u)}{\sqrt{\cosh^2(u) -
\cos^2(v)}} &=
\lim\limits_{u\to\infty}\frac{\sinh(u)}{\cosh(u)}\\
&= \lim\limits_{u\to\infty}\frac{e^u - e^{-u}}{e^u + e^{-u}}\\
&= \lim\limits_{u\to\infty}\frac{e^{2u} - 1}{e^{2u} + 1}\\
&= \lim\limits_{u\to\infty}\frac{e^{2u}}{e^{2u}}\\
&= 1\\
\lim\limits_{u\to\infty}\frac{\cosh(u)}{\sqrt{\cosh^2(u) -
\cos^2(v)}} &=
\lim\limits_{u\to\infty}\frac{\cosh(u)}{\cosh(u)}\\
&= 1
\end{align*}

 

[jvicens]

Therefore, as \(u\to\infty\), \(\mathbf{u}_1\sim\cos(v)\unit{i} + \sin(v)\unit{j}\). Similarly,

\begin{align*}
\lim\limits_{u\to\infty}\frac{-\cosh(u)}{\sqrt{\cosh^2(u) -
\cos^2(v)}} &=
\lim\limits_{u\to\infty}\frac{-\cosh(u)}{\cosh(u)}\\
&= \lim\limits_{u\to\infty}-1\\
&= -1\\
\lim\limits_{u\to\infty}\frac{\sinh(u)}{\sqrt{\cosh^2(u) -
\cos^2(v)}} &=
\lim\limits_{u\to\infty}\frac{\sinh(u)}{\cosh(u)}\\
&= 1
\end{align*}

Therefore, as \(u\to\infty\), \(\mathbf{u}_2\sim -\sin(v)\unit{i} + \cos(v)\unit{j}\). This approach is valid because as \(u\) approaches infinity, the hyperbolic functions will dominate over the trigonometric functions, so their values can be approximated by just considering the hyperbolic terms. This allows for a simpler and more accurate approximation of the vectors \(\mathbf{u}_1\) and \(\mathbf{u}_2\).