Dirty Limits

Calculus Level 5

\[\lim _{ x\rightarrow \infty }{ \frac { \ln { \left( x+\ln { \left( x+\ln { \left( { x }^{ \frac { 1 }{ x } } \right) } \right) } \right) } }{ \log _{ 10 }{ \left( { x }^{ 2 } \right) } } } \]

This expression can be expressed as \[\frac { A }{ B } \left( \ln { \left( C \right) } +\ln { \left( D \right) } \right) \] for co-prime positive integers \(A\) and \(B\), and prime numbers \(C\) and \(D\).

Find \[A+B+C+D\]

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