What would be limit of $$\cos(kt+kux)$$ as $k\to0$, where $$k=2\sqrt{1-a^{2}}$$ $$u=b\left(6-k^{2}\right)$$ and $a$ and $b$ are real numbers?
The answer that I have found in the book is $$\lim_{k\rightarrow 0}\cos(kt+kux)=(t+6xb)^{2}$$
I have tried it many times, but can't do it. Can anyone help solve it or give a hint?
For small $z$ $\cos(z)\approx1-\frac{z^2}{2}$. For small $k$, $u\approx 6b$. Then $$\cos(k(x+6bt))\approx 1-\frac{k^2}{2}(t+6xb)^2$$