The discussion revolves around evaluating the limit of the expression \(\lim_{x\to1}\frac{\sin(x^{3}-1)}{x-1}\) without using L'Hôpital's rule. Participants explore various approaches to solve the limit, focusing on algebraic manipulation and trigonometric identities.
Participants do not reach a consensus on how to evaluate the limit without L'Hôpital's rule, and multiple approaches are presented without agreement on a definitive solution.
Participants express uncertainty about the steps involved in evaluating the limit, particularly regarding the behavior of the sine function and the polynomial as \(x\) approaches 1. There are unresolved mathematical steps and dependencies on the manipulation of the limit expression.
This discussion may be of interest to students or individuals seeking alternative methods for evaluating limits in calculus, particularly those looking to avoid L'Hôpital's rule.
Yankel said:Hello,
I need some help with this limit, I have no clue how to do this with the next constraint: No use of L'hopital rule...
Thank you !
\lim_{x\to1}\frac{sin(x^{3}-1)}{x-1}
chisigma said:Try multiplying numerator and denominator by $x^{2} + x + 1$...
Kind regards
$\chi$ $\sigma$
Yankel said:I thought about it, it gives me what's in the sine, however x->1 and not x->0, so I am still stuck...