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lim-x-0-tan-tanx-sin-1-cosx-




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

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