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Question Number 222 by 123456 last updated on 25/Jan/15
lim_(x→0) ((sin x)/(arcsin x))
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:{x}}{\mathrm{arcsin}\:{x}} \\ $$
Answered by ghosea last updated on 16/Dec/14
lim_(x→0) ((sin x)/(arcsin x)) =lim_(x→0) ((sin x)/x)∙(x/(arcsin x)) =lim_(x→0) ((sin x)/x)∙lim_(x→0) (x/(arcsin x)) =1∙1=1
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{x}}{\mathrm{arcsin}\:{x}} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{x}}{{x}}\centerdot\frac{{x}}{\mathrm{arcsin}\:{x}} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{x}}{{x}}\centerdot\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}}{\mathrm{arcsin}\:{x}} \\ $$$$=\mathrm{1}\centerdot\mathrm{1}=\mathrm{1} \\ $$

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