Write down the expansion of \(\ln ( 1 + 2 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\).
Hence, or otherwise, find the first two non-zero terms in the expansion of
$$\ln [ ( 1 + 2 x ) ( 1 - 2 x ) ]$$
in ascending powers of \(x\) and state the range of values of \(x\) for which the expansion is valid.
Find \(\lim _ { x \rightarrow 0 } \left[ \frac { 3 x - x \sqrt { 9 + x } } { \ln [ ( 1 + 2 x ) ( 1 - 2 x ) ] } \right]\).