Inverted Integrated Limits

Calculus Level 4

Let \(\displaystyle { a }_{ n }=\int _{ \frac { 1 }{ n+1 } }^{ \frac { 1 }{ n } }{ \tan ^{ -1 }{ (nx) } } dx\) and \( \displaystyle { b }_{ n }=\int _{ \frac { 1 }{ n+1 } }^{ \frac { 1 }{ n } }{ \sin ^{ -1 }{ (nx) } } dx\), then \(\displaystyle \lim _{ n\rightarrow \infty }{ \frac { { a }_{ n } }{ { b }_{ n } } }\) has the value equal to

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