Part 4/4:
Therefore, the limit of our original function as ( x ) approaches zero is:
[
\lim_{{x \to 0}} \frac{\sin(tan(x)) - tan(sin(x))}{\sin^{-1}(tan(x)) - tan^{-1}(sin(x))} = 1
]
In conclusion, L'Hôpital's Rule proves to be an incredibly useful tool in calculus for evaluating limits, especially when faced with indeterminate forms. The sequential application of derivatives can ultimately simplify complex problems, leading to insightful solutions, as demonstrated in this evaluation.