Part 1/4:

Exploring Limits Using L'Hôpital's Rule

Introduction

In calculus, the evaluation of limits can often lead to indeterminate forms. One such prominent method to simplify these calculations is L'Hôpital's Rule. This article will explore a specific example of applying L'Hôpital's Rule to evaluate the limit as ( x ) approaches zero for a particular function. By utilizing derivatives and computational tools, we will uncover the outcome of this limit.

Problem Statement

For our analysis, we want to find the limit:

[

\lim_{{x \to 0}} \frac{{\sin(\tan(x)) - \tan(\sin(x))}}{{\sin^{-1}(\tan(x)) - \tan^{-1}(\sin(x))}}

]

As ( x ) approaches zero, both the numerator and the denominator approach zero, leading us into an indeterminate form of ( 0/0 ).