# 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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