Prove this equality \lim_{x\to0}(\frac{\cos(\pi x)}{\sin(\pi x)}-\frac{\pi x}{\sin^2(\pi x)})=0

Sarah-Louise Prince

Sarah-Louise Prince

Answered question

2022-02-25

Prove this equality
limx0(cos(πx)sin(πx)πxsin2(πx))=0

Answer & Explanation

mapogoosae

mapogoosae

Beginner2022-02-26Added 4 answers

Using l’Hospital’s rule:
limx0(cosπxsinπxπxsin2πx)=limx0sinπxcosπxπxsin2πx
=limx0sin2πx2πx2sin2πx
=limx02πcos2πx2π4πsinπxcosπx
=limx0cos2πx1sin2πx
=limx02πsin2πx2πcosπx
=0

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