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please-prove-that-lim-x-0-x-sinx-x-3-1-6-by-using-x-3y-and-sin3y-3siny-4sin-3-y-




Question Number 71362 by malwaan last updated on 14/Oct/19
please prove that  lim_(x→0) ((x−sinx)/x^3 ) =(1/6) by using  x=3y and   sin3y=3siny−4sin^3 y
$$\boldsymbol{\mathrm{please}}\:\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\frac{\boldsymbol{{x}}−\boldsymbol{{sinx}}}{\boldsymbol{{x}}^{\mathrm{3}} }\:=\frac{\mathrm{1}}{\mathrm{6}}\:\boldsymbol{{by}}\:\boldsymbol{{using}} \\ $$$$\boldsymbol{{x}}=\mathrm{3}\boldsymbol{{y}}\:\boldsymbol{{and}}\: \\ $$$$\boldsymbol{{sin}}\mathrm{3}\boldsymbol{{y}}=\mathrm{3}\boldsymbol{{siny}}−\mathrm{4}\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{y}} \\ $$

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