Limits and Integral of transcendental function

Calculus Level pending

This is graph of function \({ f }_{ (x) }={ e }^{ 2x }\) and its inverse function \(y={ g }_{ (x) }\) (where e is base of natural logarithm)

(1) Solve \(\lim _{ x\rightarrow 0 }{ \frac { { f }_{ (2x) }-1 }{ { g }_{ (2x+1) } } }\)

(2) And, consider a line that passes through a point \(({ e }^{ 4 },{ g }_{ ({ e }^{ 4 }) })\) and is parallel with x axis. Let a point where this line intersects with curve \(y={ f }_{ (x) }\) be \((a,{ f }_{ (a) })\). Calculate area that is surrounded by x axis, y axis, curve \(y={ f }_{ (x) }\), and line \(x=a\).

Let the answer of question (1) be X and the answer of question (2) be Y. Evaluate \(\boxed{XY}\).

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