Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
Given that \(y = \sqrt { 3 + \mathrm { e } ^ { x } }\), find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 0\).
Using Maclaurin's theorem, show that, for small values of \(x\),
$$\sqrt { 3 + \mathrm { e } ^ { x } } \approx 2 + \frac { 1 } { 4 } x + \frac { 7 } { 64 } x ^ { 2 }$$
Find
$$\lim _ { x \rightarrow 0 } \left[ \frac { \sqrt { 3 + \mathrm { e } ^ { x } } - 2 } { \sin 2 x } \right]$$