Question Number 194640 by cortano12 last updated on 12/Jul/23
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Answered by horsebrand11 last updated on 12/Jul/23
$$\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}}}\:=\:\mathrm{y} \\ $$$$\:\underset{\mathrm{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{y}}{\mathrm{y}\:\mathrm{sin}\:\mathrm{3y}}\:=\underset{\mathrm{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} \frac{\mathrm{1}}{\mathrm{2}}\mathrm{y}}{\mathrm{2y}\:\mathrm{sin}\:\mathrm{3y}} \\ $$$$\:=\:\frac{\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\mathrm{6}}=\frac{\mathrm{1}}{\mathrm{24}} \\ $$