Final answer:
The student is required to find the limit of a function as x approaches 0. Since direct substitution leads to an indeterminate form, l'Hospital's Rule should be employed, which involves differentiating the numerator and the denominator separately to find the limit.
Step-by-step explanation:
The student is asking to find the limit of the function (e²x − e⁻²x − 4x) / (x − sin x) as x approaches 0. To solve this problem, we can initially try to apply elementary limit laws and subsequently use l'Hospital's Rule if necessary. When we substitute x = 0 directly into the function, we get an indeterminate form (0/0), which suggests that l'Hospital's Rule might be an appropriate method to use.
However, before a direct application of l'Hospital's Rule, it is useful to see if the function can be simplified or if the limit can be evaluated using standard limit results. In this case, both the numerator and the denominator approach zero as x approaches zero, so l'Hospital's Rule is indeed the proper approach. After applying l'Hospital's Rule by differentiating the numerator and denominator separately and then taking the limit, we find the answer.
It is important to recognize that this method is applicable here due to the indeterminate form we encountered. This limit computation is an example of the intricate procedures in calculus that help us understand the behavior of functions near points where they are not immediately defined.