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Question Number 160912 by cortano last updated on 09/Dec/21
lim_(x→0) ((tan (x+2)tan (2−x)−tan^2 (2))/(3x tan x)) =?
$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\left(\mathrm{x}+\mathrm{2}\right)\mathrm{tan}\:\left(\mathrm{2}−\mathrm{x}\right)−\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{2}\right)}{\mathrm{3x}\:\mathrm{tan}\:\mathrm{x}}\:=? \\ $$
Answered by blackmamba last updated on 09/Dec/21
{ ((tan (x+2)=((tan 2+tan x)/(1−tan 2 tan x)))),((tan (2−x)=((tan 2−tan x)/(1+tan 2 tan x)))),((tan (x+2)tan (2−x)=((tan^2 (2)−tan^2 (x))/(1−tan^2 (2)tan^2 (x))))) :} lim_(x→0) ((((tan^2 (2)−tan^2 (x))/(1−tan^2 (2) tan^2 (x))) −tan^2 (2))/(3x tan x)) = lim_(x→0) ((−tan^2 (x)+tan^2 (x)tan^4 (2))/(3x tan x (1−tan^2 (2)tan^2 (x)))) = lim_(x→0) ((tan^2 (x))/(3x tan x)) . lim_(x→0) (((tan^4 (2)−1)/(1−tan^2 (2) tan^2 (x)))) = ((tan^4 (2)−1)/3)
$$\:\begin{cases}{\mathrm{tan}\:\left({x}+\mathrm{2}\right)=\frac{\mathrm{tan}\:\mathrm{2}+\mathrm{tan}\:{x}}{\mathrm{1}−\mathrm{tan}\:\mathrm{2}\:\mathrm{tan}\:{x}}}\\{\mathrm{tan}\:\left(\mathrm{2}−{x}\right)=\frac{\mathrm{tan}\:\mathrm{2}−\mathrm{tan}\:{x}}{\mathrm{1}+\mathrm{tan}\:\mathrm{2}\:\mathrm{tan}\:{x}}}\\{\mathrm{tan}\:\left({x}+\mathrm{2}\right)\mathrm{tan}\:\left(\mathrm{2}−{x}\right)=\frac{\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{2}\right)−\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)}{\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{2}\right)\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)}}\end{cases} \\ $$$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{2}\right)−\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)}{\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{2}\right)\:\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)}\:−\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{2}\right)}{\mathrm{3}{x}\:\mathrm{tan}\:{x}}\: \\ $$$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)+\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)\mathrm{tan}\:^{\mathrm{4}} \left(\mathrm{2}\right)}{\mathrm{3}{x}\:\mathrm{tan}\:{x}\:\left(\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{2}\right)\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)\right)} \\ $$$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)}{\mathrm{3}{x}\:\mathrm{tan}\:{x}}\:.\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{tan}\:^{\mathrm{4}} \left(\mathrm{2}\right)−\mathrm{1}}{\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \left(\mathrm{2}\right)\:\mathrm{tan}\:^{\mathrm{2}} \left({x}\right)}\right) \\ $$$$\:=\:\frac{\mathrm{tan}\:^{\mathrm{4}} \left(\mathrm{2}\right)−\mathrm{1}}{\mathrm{3}} \\ $$

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