A blend of Graphs and limits!

Calculus Level 5

The area bounded by the points lying in between the lines \(x+5=0\) and \(x-5=0\) satisfying the equation \(\left| \left\lfloor y \right\rfloor \right| =\left| \left\lfloor \tan ^{ -1 }{ \left\lfloor x \right\rfloor } \right\rfloor \right| \) is A.

Consider a function \( f:R\rightarrow R\) . The graph of the above function has an asymptote \(2y=x+2\) . Then \(\lim _{ x\rightarrow \infty }{ { \left[ \frac { { \left[ \left( 2f\left( x \right) +1 \right) \left( 2f\left( x \right) +3 \right) \left( 2f\left( x \right) -5 \right) \left( 2f\left( x \right) -3 \right) \left( 2f\left( x \right) -1 \right) \right] }^{ \frac { 1 }{ 5 } }-f\left( x \right) }{ f\left( x \right) } \right] }^{ x\left( 2f\left( x \right) -x \right) } } =\quad { e }^{ L }\) . Then \(\left| A\times L \right| =\)

Details and assumptions :

\(\left| . \right| \) represents Absolute Value function .

\(\left\lfloor . \right\rfloor \) represents Greatest Integer function .

'e' represents exponential number .

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