My Cousin's Challenge: Solve a Tough Limit Problem

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SUMMARY

The limit problem presented involves computing the limit as x approaches 0 for the expression (sin(tan(x)) - tan(sin(x))) / (arcsin(arctan(x)) - arctan(arcsin(x))). The discussion highlights the use of L'Hôpital's rule, which proved ineffective for this problem. Instead, the recommended approach is to expand both the numerator and the denominator using Taylor series up to the first significant term to simplify the computation.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with Taylor series expansions
  • Knowledge of L'Hôpital's rule
  • Basic trigonometric functions and their properties
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  • Review the application of L'Hôpital's rule in limit problems
  • Explore advanced limit techniques in calculus
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My cousin challenges me with a tough limit problem.

Compute the limit as x->0 of (sin(tanx)-tan(sinx))/(arcsin(arctanx)-arctan(arcsinx)).

This limit seems impossible to do.

I used L'Hopital's rule but it didn't help at all.

Can anyone find a way to do it?
 
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Expand the numerator and the denominator as Taylor series up to the first significant term.
 

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