Without using L'Hospital's rule, what is \lim_{x \to 0} \frac{\tan

hadejada7x

hadejada7x

Answered question

2021-12-30

Without using LHospitals rule, what is limx0tanxsinxx3?

Answer & Explanation

Becky Harrison

Becky Harrison

Beginner2021-12-31Added 40 answers

I would do it this way:
tanxsinxx3=sinx(1cosx)x3cosx=sinx(1cos2x)x3cosx(1+cosx)=sin3xx31cosx(1+cosx)
Added: it may be shortened, using the result of this standard high-school exercise:
limx01cosxx2=12
Corgnatiui

Corgnatiui

Beginner2022-01-01Added 35 answers

1cosx=2sin2(x2) , so you need x2 in the denominator to have the pairing.
Vasquez

Vasquez

Expert2022-01-08Added 669 answers

limx0tanxsinxx3==limx0tanxxlimx01cosxx2==12limx0sin2x2(x2)2

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