Question-85542

Question Number 85542 by TawaTawa1 last updated on 22/Mar/20

Commented by TawaTawa1 last updated on 22/Mar/20

Evaluate: lim_(x→(x/2)) (x − (π/2)) tan x

$$\mathrm{Evaluate}:\:\:\:\:\:\underset{{x}\rightarrow\frac{\mathrm{x}}{\mathrm{2}}} {\mathrm{lim}}\:\:\left(\mathrm{x}\:\:−\:\:\frac{\pi}{\mathrm{2}}\right)\:\mathrm{tan}\:\mathrm{x} \\ $$

Commented by mathmax by abdo last updated on 22/Mar/20

let f(x)=(x−(π/2))tanx changement x−(π/2)=t give f(x)=t tan((π/2)+t) =t((sin((π/2)+t))/(cos((π/2)+t))) =t ((cost)/(−sint)) =−(t/(tant)) lim_(x→(π/2)) f(x)=lim_(t→0) −(1/((((tant)/t)))) =−1

$${let}\:{f}\left({x}\right)=\left({x}−\frac{\pi}{\mathrm{2}}\right){tanx}\:\:{changement}\:{x}−\frac{\pi}{\mathrm{2}}={t}\:{give} \\ $$$${f}\left({x}\right)={t}\:{tan}\left(\frac{\pi}{\mathrm{2}}+{t}\right)\:={t}\frac{{sin}\left(\frac{\pi}{\mathrm{2}}+{t}\right)}{{cos}\left(\frac{\pi}{\mathrm{2}}+{t}\right)}\:={t}\:\frac{{cost}}{−{sint}}\:=−\frac{{t}}{{tant}} \\ $$$${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \:{f}\left({x}\right)={lim}_{{t}\rightarrow\mathrm{0}} \:\:−\frac{\mathrm{1}}{\left(\frac{{tant}}{{t}}\right)}\:=−\mathrm{1} \\ $$

Commented by TawaTawa1 last updated on 22/Mar/20

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

Commented by turbo msup by abdo last updated on 23/Mar/20

$${you}\:{are}\:{welcome}\:{miss}\:{tawa} \\ $$

Answered by jagoll last updated on 22/Mar/20

lim_(x→(π/2)) (x−(π/2)) cot ((π/2)−x) lim_(t→0) ((−t)/(tan t)) = −1

$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\left(\mathrm{x}−\frac{\pi}{\mathrm{2}}\right)\:\mathrm{cot}\:\left(\frac{\pi}{\mathrm{2}}−\mathrm{x}\right) \\ $$$$\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{t}}{\mathrm{tan}\:\mathrm{t}}\:=\:−\mathrm{1} \\ $$

Commented by TawaTawa1 last updated on 22/Mar/20

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

Commented by TawaTawa1 last updated on 22/Mar/20

$$\mathrm{Sir},\:\mathrm{help}\:\mathrm{me}\:\mathrm{solve}\:\mathrm{question}\:\:\mathrm{85546} \\ $$

Commented by jagoll last updated on 23/Mar/20

writting is too small, i did not clearly read it

$$\mathrm{writting}\:\mathrm{is}\:\mathrm{too}\:\mathrm{small},\:\mathrm{i}\:\mathrm{did}\:\mathrm{not} \\ $$$$\mathrm{clearly}\:\mathrm{read}\:\mathrm{it} \\ $$

Commented by TawaTawa1 last updated on 23/Mar/20

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir},\:\mathrm{please}\:\mathrm{check},\:\mathrm{i}\:\mathrm{retype}. \\ $$

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