calculate-lim-x-0-tan-2x-2tanx-2tan-3-x-x-5-

Question Number 81027 by abdomathmax last updated on 09/Feb/20

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{tan}\left(\mathrm{2}{x}\right)−\mathrm{2}{tanx}−\mathrm{2}{tan}^{\mathrm{3}} {x}}{{x}^{\mathrm{5}} } \\ $$

Commented by john santu last updated on 09/Feb/20

lim_(x→0) ((((tan2x )/(2x)).(2x)−((2tan x)/x).(x)−((2tan^3 x )/x^3 ).(x^3 ))/x^5 ) = lim_(x→0) ((2x−2x−2x^3 )/(x^5 )) =lim_(x→0) ((−2)/x^2 ) = −∞ lim_(x→0^+ ) ((−1)/x^2 ) = −∞, lim_(x→0^− ) ((−1)/x^2 ) = −∞

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{tan2}{x}\:}{\mathrm{2}{x}}.\left(\mathrm{2}{x}\right)−\frac{\mathrm{2tan}\:{x}}{{x}}.\left({x}\right)−\frac{\mathrm{2tan}^{\mathrm{3}} {x}\:}{{x}^{\mathrm{3}} }.\left({x}^{\mathrm{3}} \right)}{{x}^{\mathrm{5}} }\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{x}−\mathrm{2}{x}−\mathrm{2}{x}^{\mathrm{3}} }{{x}^{\mathrm{5}} \:}\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{2}}{{x}^{\mathrm{2}} }\:=\:−\infty \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{−\mathrm{1}}{{x}^{\mathrm{2}} }\:=\:−\infty,\:\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\:\frac{−\mathrm{1}}{{x}^{\mathrm{2}} }\:=\:−\infty \\ $$

Commented by john santu last updated on 09/Feb/20

Commented by mathocean1 last updated on 09/Feb/20

please which Application do you use to make this graphic sir?

$$\mathrm{please}\:\mathrm{which}\:\mathrm{Application}\:\mathrm{do}\:\mathrm{you}\:\mathrm{use}\:\mathrm{to} \\ $$$$\mathrm{make}\:\mathrm{this}\:\mathrm{graphic}\:\mathrm{sir}? \\ $$

Commented by john santu last updated on 09/Feb/20