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Question Number 195325 Answers: 2 Comments: 0

prove that lim_(x→(π/2)) ((tan((x/2))−1)/(x−(π/2)))=1

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

Question Number 200302 Answers: 1 Comments: 0

Question Number 195320 Answers: 1 Comments: 0

I_n =∫_0 ^( +∞) t^(−2t) sin^(2n) tdt Prove that I_n =(1/(1−e^(−2π) )) ∫^( π) _( 0) e^(−2t) sin^(2n) t dt and I_n ∽ _(∞) (1/(2sh(π)))(√(π/n))

$$\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\:+\infty} {t}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} {tdt} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{I}_{\mathrm{n}} =\frac{\mathrm{1}}{\mathrm{1}−{e}^{−\mathrm{2}\pi} }\:\:\underset{\:\mathrm{0}} {\int}^{\:\pi} {e}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} {t}\:{dt} \\ $$$$\mathrm{and}\:\:\mathrm{I}_{\mathrm{n}} \underset{\infty} {\:\:\backsim\:\:}\:\frac{\mathrm{1}}{\mathrm{2}{sh}\left(\pi\right)}\sqrt{\frac{\pi}{{n}}} \\ $$

Question Number 195342 Answers: 4 Comments: 0

1. Prove that ∀n ∈ N^∗ , 4^n (n!)^3 < (n+1)^(3n) . 2. Solve the equations in Z^2 : a./ 2x^3 +xy−7=0 , b./ x(x+1)(x+7)(x+8)=y^2 .

$$\:\:\mathrm{1}.\:\mathrm{Prove}\:\mathrm{that}\:\:\forall{n}\:\in\:\mathbb{N}^{\ast} \:,\:\mathrm{4}^{{n}} \left({n}!\right)^{\mathrm{3}} \:<\:\left({n}+\mathrm{1}\right)^{\mathrm{3}{n}} \:. \\ $$$$\mathrm{2}.\:\mathrm{Solve}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{in}\:\mathbb{Z}^{\mathrm{2}} \:: \\ $$$$\:\:\:\:\:{a}./\:\:\mathrm{2}{x}^{\mathrm{3}} +{xy}−\mathrm{7}=\mathrm{0}\:, \\ $$$$\:\:\:\:\:{b}./\:\:{x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{7}\right)\left({x}+\mathrm{8}\right)={y}^{\mathrm{2}} . \\ $$

Question Number 195341 Answers: 0 Comments: 0

Question Number 195315 Answers: 1 Comments: 0

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Question Number 195555 Answers: 1 Comments: 0

Question Number 195554 Answers: 0 Comments: 0

Question Number 195553 Answers: 1 Comments: 0

Question Number 195301 Answers: 1 Comments: 0

{ ((x^2 +y=11)),((x+y^2 =7)) :}⇒ x,y=?

$$\begin{cases}{{x}^{\mathrm{2}} +{y}=\mathrm{11}}\\{{x}+{y}^{\mathrm{2}} =\mathrm{7}}\end{cases}\Rightarrow\:{x},{y}=? \\ $$

Question Number 195292 Answers: 1 Comments: 0

Question Number 195291 Answers: 1 Comments: 0

it is given a,b,c ∈ N^∗ and ab<c . Prove that a+b≤c.

$$\:\:{it}\:{is}\:{given}\:{a},{b},{c}\:\in\:\mathbb{N}^{\ast} \:\:{and}\:\:{ab}<{c}\:.\:{Prove}\:{that}\:{a}+{b}\leqslant{c}. \\ $$

Question Number 195288 Answers: 0 Comments: 1

x, y, z∈R_+ , P = (x/(x + y)) + (y/(y + z)) + (z/(z + x)), Q = (y/(x + y)) + (z/(y + z)) + (x/(z + x)), Q = (z/(x + y)) + (x/(y + z)) + (y/(z + x)). f(x, y, z)=max{P, Q, R}, find f_(min) .

$${x},\:{y},\:{z}\in\mathbb{R}_{+} , \\ $$$${P}\:=\:\frac{{x}}{{x}\:+\:{y}}\:+\:\frac{{y}}{{y}\:+\:{z}}\:+\:\frac{{z}}{{z}\:+\:{x}}, \\ $$$${Q}\:=\:\frac{{y}}{{x}\:+\:{y}}\:+\:\frac{{z}}{{y}\:+\:{z}}\:+\:\frac{{x}}{{z}\:+\:{x}}, \\ $$$${Q}\:=\:\frac{{z}}{{x}\:+\:{y}}\:+\:\frac{{x}}{{y}\:+\:{z}}\:+\:\frac{{y}}{{z}\:+\:{x}}. \\ $$$${f}\left({x},\:{y},\:{z}\right)=\mathrm{max}\left\{{P},\:{Q},\:{R}\right\},\:\mathrm{find}\:{f}_{\mathrm{min}} . \\ $$

Question Number 195287 Answers: 2 Comments: 0

Question Number 200309 Answers: 2 Comments: 0

Question Number 195289 Answers: 1 Comments: 0

Question Number 195290 Answers: 1 Comments: 0

A professor said 0∣0 because 0= 0×a+0 , a∈ N. Can you prove?

$$\:{A}\:{professor}\:{said}\:\:\mathrm{0}\mid\mathrm{0}\:{because}\:\mathrm{0}=\:\mathrm{0}×{a}+\mathrm{0}\:\:\:,\:{a}\in\:\mathbb{N}.\:{Can}\:{you}\:{prove}? \\ $$

Question Number 195322 Answers: 1 Comments: 0

Question Number 200300 Answers: 1 Comments: 0

Question Number 200299 Answers: 1 Comments: 0

Question Number 200298 Answers: 1 Comments: 1

Question Number 195273 Answers: 0 Comments: 1

Question Number 195454 Answers: 4 Comments: 1

Question Number 195453 Answers: 1 Comments: 1

lim_(x→+∞ ) x ln(((e^x + 1)/(e^x −1))) ?

$$\:\:\:\:\mathrm{lim}_{{x}\rightarrow+\infty\:\:} \:{x}\:{ln}\left(\frac{{e}^{{x}} +\:\mathrm{1}}{{e}^{{x}} −\mathrm{1}}\right)\:\:? \\ $$

Question Number 195263 Answers: 2 Comments: 0

{ ((y sin x+cos x = 2)),((4sin x−2y cos x = y)) :} tan x =?

$$\:\:\:\:\:\:\begin{cases}{{y}\:\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\:=\:\mathrm{2}}\\{\mathrm{4sin}\:{x}−\mathrm{2}{y}\:\mathrm{cos}\:{x}\:=\:{y}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\mathrm{tan}\:{x}\:=?\: \\ $$

Question Number 195259 Answers: 0 Comments: 1

prove that lim_(n→∞) ((e^n ∙(n!))/(n^n (√n)))=(√(2π))

$${prove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{e}^{{n}} \centerdot\left({n}!\right)}{{n}^{{n}} \:\sqrt{{n}}}=\sqrt{\mathrm{2}\pi} \\ $$

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