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Question Number 28095 by tawa tawa last updated on 20/Jan/18

lim_(x→0^− )   (1 + tanx)^(−cotx)

$$\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\:\:\left(\mathrm{1}\:+\:\mathrm{tanx}\right)^{−\mathrm{cotx}} \\ $$

Commented by abdo imad last updated on 20/Jan/18

= lim_(x→0^− )   e^(−cotanx ln(1+tanx))      but   ln (1+tanx) _(x∈V(0)) ∼  tanx  ⇒ −cotanx ln(1+tanx)  ∼ −((tanx)/(tanx))=−1  ⇒ lim_(x→0^(− ) )   (1+tanx)^(−cotanx)  =(1/e) .

$$=\:{lim}_{{x}\rightarrow\mathrm{0}^{−} } \:\:{e}^{−{cotanx}\:{ln}\left(\mathrm{1}+{tanx}\right)} \:\:\:\:\:{but}\: \\ $$$${ln}\:\left(\mathrm{1}+{tanx}\right)\:_{{x}\in{V}\left(\mathrm{0}\right)} \sim\:\:{tanx}\:\:\Rightarrow\:−{cotanx}\:{ln}\left(\mathrm{1}+{tanx}\right) \\ $$$$\sim\:−\frac{{tanx}}{{tanx}}=−\mathrm{1}\:\:\Rightarrow\:{lim}_{{x}\rightarrow\mathrm{0}^{−\:} } \:\:\left(\mathrm{1}+{tanx}\right)^{−{cotanx}} \:=\frac{\mathrm{1}}{{e}}\:. \\ $$

Commented by tawa tawa last updated on 21/Jan/18

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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