Question Number 93824 by student work last updated on 15/May/20
Answered by john santu last updated on 15/May/20
$$\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{lny}\:=\:\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{tan}\:\left(\mathrm{3t}+\frac{\pi}{\mathrm{2}}\right)\mathrm{ln}\left(\mathrm{tan}\:\left(\frac{\mathrm{3t}}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}\right)\right) \\ $$$$=\:\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\left(\mathrm{tan}\:\left(\frac{\mathrm{3t}}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}\right)\right)}{\mathrm{cot}\:\left(\mathrm{3t}+\frac{\pi}{\mathrm{2}}\right)} \\ $$$$=\:\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\left(\mathrm{tan}\:\left(\frac{\mathrm{3t}}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}\right)\right)}{−\mathrm{tan}\:\mathrm{3t}} \\ $$$$\mathrm{now}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{solve}\: \\ $$