# Limit of \cos(kt+kux) as k \to 0 where k=2\sqrt{1-a^2} u=b(6-k^2) and a and b

Limit of $\mathrm{cos}\left(kt+kux\right)$ as $k\to 0$
where
$k=2\sqrt{1-{a}^{2}}$
$u=b\left(6-{k}^{2}\right)$
and a and b are real numbers?
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Olive Guzman
For small $\mathrm{cos}\left(z\right)\approx 1-\frac{{z}^{2}}{2}$. For small $k,u\approx 6b$. Then
$\mathrm{cos}\left(k\left(x+6bt\right)\right)\approx 1-\frac{{k}^{2}}{2}{\left(t+6xb\right)}^{2}$