Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 122835 by liberty last updated on 20/Nov/20

lim_(x→0) (1/x) ((1/(sin x)) − (3/(sin 3x)) ) =?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\:\left(\frac{\mathrm{1}}{\mathrm{sin}\:{x}}\:−\:\frac{\mathrm{3}}{\mathrm{sin}\:\mathrm{3}{x}}\:\right)\:=? \\ $$

Answered by $@y@m last updated on 20/Nov/20

lim_(x→0) (1/x) (((sin 3x−3sin x)/(sin x.sin 3x)) ) lim_(x→0) (1/x) (((−4sin^3 x)/(sin x.sin 3x)) ) lim_(x→0) (1/x) (((−4sin^2 x)/(sin 3x)) ) (−4)lim_(x→0) ((sin^2 x)/x^2 ) ÷ (lim_(x→0) ((sin 3x)/(3x)) ×3) =((−4)/3)

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\:\left(\frac{\mathrm{sin}\:\mathrm{3}{x}−\mathrm{3sin}\:{x}}{\mathrm{sin}\:{x}.\mathrm{sin}\:\mathrm{3}{x}}\:\right)\: \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\:\left(\frac{−\mathrm{4sin}\:^{\mathrm{3}} {x}}{\mathrm{sin}\:{x}.\mathrm{sin}\:\mathrm{3}{x}}\:\right)\: \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\:\left(\frac{−\mathrm{4sin}^{\mathrm{2}} {x}}{\mathrm{sin}\:\mathrm{3}{x}}\:\right)\: \\ $$$$\:\left(−\mathrm{4}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} }\:\boldsymbol{\div}\:\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{sin}\:\mathrm{3}{x}}{\mathrm{3}{x}}\:×\mathrm{3}\right)\: \\ $$$$=\frac{−\mathrm{4}}{\mathrm{3}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com