Question Number 205339 by depressiveshrek last updated on 17/Mar/24
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{tan}\left(\mathrm{tan}{x}\right)}{\mathrm{sin}\left(\mathrm{1}−\mathrm{cos}{x}\right)} \\ $$
Answered by MM42 last updated on 18/Mar/24
$$={lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{tan}\left({x}\right)}{{sin}\left(\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} \right)} \\ $$$$={lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}}{\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} }\: \\ $$$$={lim}_{{x}\rightarrow\mathrm{0}} \frac{\mathrm{2}}{{x}}=\begin{cases}{\:+\infty\:\:\:{if}\:\:\:\:{x}\rightarrow\mathrm{0}^{+} }\\{−\infty\:\:\:{if}\:\:\:\:{x}\rightarrow\mathrm{0}^{−} }\end{cases} \\ $$$$ \\ $$