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lim-x-0-x-3-x-sin-x-




Question Number 206069 by MathematicalUser2357 last updated on 06/Apr/24
lim_(x→0) ((−x^3 +x)/(sin x))
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−{x}^{\mathrm{3}} +{x}}{\mathrm{sin}\:{x}} \\ $$
Answered by MetaLahor1999 last updated on 06/Apr/24
lim_(x→0)  ((x−x^3 )/(sin(x)))=lim_(x→0)  ((x(1−x^2 ))/(x((sin(x))/x)))                            =lim_(x→0)  ((1−x^2 )/((sin(x))/x))                            =1.
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}−{x}^{\mathrm{3}} }{{sin}\left({x}\right)}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}{{x}\frac{{sin}\left({x}\right)}{{x}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−{x}^{\mathrm{2}} }{\frac{{sin}\left({x}\right)}{{x}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{1}. \\ $$

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