MHB Solving Trigonometric Limit w/o L'Hospital Rule

[Yankel]

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]

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}

Try multiplying numerator and denominator by $x^{2} + x + 1$...

Kind regards

$\chi$ $\sigma$

 

[Yankel]

Try multiplying numerator and denominator by $x^{2} + x + 1$...

Kind regards

$\chi$ $\sigma$

I thought about it, it gives me what's in the sine, however x->1 and not x->0, so I am still stuck...

 

[chisigma]

I thought about it, it gives me what's in the sine, however x->1 and not x->0, so I am still stuck...

$\displaystyle \lim_{x \rightarrow 1} \frac{\sin (x^{3}-1)}{x-1} = \lim_{x \rightarrow 1} \frac{\sin (x^{3}-1)}{x^{3}-1}\ (x^{2} + x + 1)$

... and now what is $\displaystyle \lim_{x \rightarrow 1} \frac{\sin (x^{3}-1)}{x^{3}-1}$?...

... and what is $\displaystyle \lim_{x \rightarrow 1} x^{2} + x + 1$?...

Kind regards

$\chi$ $\sigma$

 

[MarkFL]

Also, L'Hôpital's rule allows us to write:

$\displaystyle \lim_{x\to1}\frac{\sin(x^3-1)}{x-1}=3\lim_{x\to1}x^2\cos(x^3-1)$