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Question Number 107975    Answers: 3   Comments: 1

((♣JS♥)/(•≡•)) (1) lim_(x→0) ((sin (tan x)−tan (sin x))/(x−sin x )) (2)lim_(x→∞) x^2 (√((1−cos ((2/x)))(√((1−cos ((2/x)))(√((1−cos ((2/x)))(√(...)))))))) ?

$$\:\:\:\:\:\:\frac{\clubsuit{JS}\heartsuit}{\bullet\equiv\bullet} \\ $$$$\:\left(\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left(\mathrm{tan}\:{x}\right)−\mathrm{tan}\:\left(\mathrm{sin}\:{x}\right)}{{x}−\mathrm{sin}\:{x}\:} \\ $$$$\:\left(\mathrm{2}\right)\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{2}} \sqrt{\left(\mathrm{1}−\mathrm{cos}\:\left(\frac{\mathrm{2}}{{x}}\right)\right)\sqrt{\left(\mathrm{1}−\mathrm{cos}\:\left(\frac{\mathrm{2}}{{x}}\right)\right)\sqrt{\left(\mathrm{1}−\mathrm{cos}\:\left(\frac{\mathrm{2}}{{x}}\right)\right)\sqrt{...}}}}\:?\: \\ $$$$ \\ $$

Question Number 107974    Answers: 1   Comments: 0

Σ_(k=1) ^n (√(1+(1/k^2 )+(1/((1+k)^2 ))))

$$\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{k}^{\mathrm{2}} }+\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{k}\right)^{\mathrm{2}} }} \\ $$

Question Number 107871    Answers: 1   Comments: 1

((✓JS✓)/♥) lim_(x→0) (√((x tan x)/(sin 2x−cos 2x +1))) ?

$$\:\:\:\:\:\:\:\frac{\checkmark\mathcal{JS}\checkmark}{\heartsuit} \\ $$$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt{\frac{{x}\:\mathrm{tan}\:{x}}{\mathrm{sin}\:\mathrm{2}{x}−\mathrm{cos}\:\mathrm{2}{x}\:+\mathrm{1}}}\:?\: \\ $$

Question Number 107965    Answers: 2   Comments: 0

((⊚BeMath⊚)/) ∫ x (√(x/(2a−x))) dx ?

$$\:\:\:\:\frac{\circledcirc\mathcal{B}{e}\mathcal{M}{ath}\circledcirc}{} \\ $$$$\int\:{x}\:\sqrt{\frac{{x}}{\mathrm{2}{a}−{x}}}\:{dx}\:?\: \\ $$

Question Number 107947    Answers: 1   Comments: 0

Question Number 107946    Answers: 0   Comments: 0

Let a sequence {a_n } satisfies a_n = { ((2, n=1)),((2ln(a_(n−1) )+(1/a_(n−1) ) , n≥2)) :} Prove that a_n ≥1+(1/n) for all n∈N.

$$\mathrm{Let}\:\mathrm{a}\:\mathrm{sequence}\:\left\{{a}_{\mathrm{n}} \right\}\:\mathrm{satisfies} \\ $$$${a}_{\mathrm{n}} =\begin{cases}{\mathrm{2},\:\mathrm{n}=\mathrm{1}}\\{\mathrm{2ln}\left({a}_{\mathrm{n}−\mathrm{1}} \right)+\frac{\mathrm{1}}{{a}_{\mathrm{n}−\mathrm{1}} }\:,\:\mathrm{n}\geqslant\mathrm{2}}\end{cases} \\ $$$$\mathrm{Prove}\:\mathrm{that}\: \\ $$$${a}_{\mathrm{n}} \geqslant\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}\:\mathrm{for}\:\mathrm{all}\:\mathrm{n}\in\mathbb{N}. \\ $$

Question Number 107945    Answers: 3   Comments: 0

((○BeMath○)/(∧⌣∧)) { ((x^4 +(1/x^4 ) = 23)),((x^3 −(1/x^3 ) = ?)) :}

$$\:\:\:\:\:\:\frac{\circ\mathbb{B}{e}\mathbb{M}{ath}\circ}{\wedge\smile\wedge} \\ $$$$\:\:\:\begin{cases}{{x}^{\mathrm{4}} +\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\:=\:\mathrm{23}}\\{{x}^{\mathrm{3}} −\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:=\:?}\end{cases} \\ $$

Question Number 107941    Answers: 0   Comments: 1

Question Number 107932    Answers: 1   Comments: 3

((BeMath)/•) Given { ((tan (x−y)=(3/4))),((tan x = 2 )) :} find tan y ?

$$\:\:\:\frac{\mathbb{B}{e}\mathbb{M}{ath}}{\bullet} \\ $$$${Given}\:\begin{cases}{\mathrm{tan}\:\left({x}−{y}\right)=\frac{\mathrm{3}}{\mathrm{4}}}\\{\mathrm{tan}\:{x}\:=\:\mathrm{2}\:}\end{cases} \\ $$$${find}\:\:\mathrm{tan}\:{y}\:? \\ $$

Question Number 107930    Answers: 3   Comments: 0

((BeMath)/(•∩•)) lim_(x→0) (sin x)^(1/(ln (√x))) ?

$$\:\:\frac{\mathbb{B}{e}\mathbb{M}{ath}}{\bullet\cap\bullet} \\ $$$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{sin}\:{x}\right)^{\frac{\mathrm{1}}{\mathrm{ln}\:\sqrt{{x}}}} \:? \\ $$

Question Number 107929    Answers: 0   Comments: 0

(1/(1+2^2 +3^3 ))+(1/(2^2 +3^3 +4^4 ))+(1/(3^3 +4^4 +5^5 ))+....

$$\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{3}} +\mathrm{4}^{\mathrm{4}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} +\mathrm{4}^{\mathrm{4}} +\mathrm{5}^{\mathrm{5}} }+.... \\ $$

Question Number 107928    Answers: 0   Comments: 0

Question Number 107927    Answers: 0   Comments: 0

Question Number 107926    Answers: 2   Comments: 0

Question Number 107925    Answers: 2   Comments: 0

Question Number 107924    Answers: 1   Comments: 0

Question Number 107923    Answers: 3   Comments: 0

((⊚BeMath⊚)/) ∫_0 ^1 x^(9/2) (1−x)^(5/2) dx ?

$$\:\:\:\:\:\frac{\circledcirc\mathbb{B}{e}\mathcal{M}{ath}\circledcirc}{} \\ $$$$\:\:\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:{x}^{\frac{\mathrm{9}}{\mathrm{2}}} \:\left(\mathrm{1}−{x}\right)^{\frac{\mathrm{5}}{\mathrm{2}}} \:{dx}\:? \\ $$

Question Number 107922    Answers: 1   Comments: 0

if: y=(x/(x^2 +1)) then find (d(√y)/d(√x)) ?

$${if}:\:{y}=\frac{{x}}{{x}^{\mathrm{2}} +\mathrm{1}}\:{then}\:{find}\:\frac{{d}\sqrt{{y}}}{{d}\sqrt{{x}}}\:? \\ $$

Question Number 107913    Answers: 0   Comments: 0

X_0 = ((X_1 .F_(y1) + X_2 .F_(y2) + .....)/(F_(y2) + F_(y2 ) + .....)) Y_0 = ((Y_1 .F_(x1) + Y_2 .F_(x2) + ......)/(F_(x1) + F_(x2) + ......)) X_0 = ((x_1 .m_1 + x_2 .m_2 + x_3 .m_3 + ....)/(m_1 + m_2 + m_3 + ......)) Y_0 = ((y_1 .m_1 + y_2 .m_2 + y_3 .m_3 + ....)/(m_1 + m_2 + m_3 + .....)) X_z = ((l_1 .x_1 + l_2 .x_2 + l_3 .x_3 + .......)/(l_1 + l_2 + l_3 + ......)) Y_z = ((l_1 .y_1 + l_2 .y_2 + l_3 .y_3 + ...... )/(l_1 + l_2 + l_3 + ......))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{X}_{\mathrm{0}} \:=\:\frac{\mathrm{X}_{\mathrm{1}} .\mathrm{F}_{\mathrm{y1}} \:+\:\mathrm{X}_{\mathrm{2}} .\mathrm{F}_{\mathrm{y2}} \:+\:.....}{\mathrm{F}_{\mathrm{y2}} \:+\:\mathrm{F}_{\mathrm{y2}\:} +\:.....} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Y}_{\mathrm{0}} \:=\:\frac{\mathrm{Y}_{\mathrm{1}} .\mathrm{F}_{\mathrm{x1}} \:+\:\mathrm{Y}_{\mathrm{2}} .\mathrm{F}_{\mathrm{x2}} \:+\:......}{\mathrm{F}_{\mathrm{x1}} \:+\:\mathrm{F}_{\mathrm{x2}} +\:......} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\mathrm{X}_{\mathrm{0}} \:=\:\frac{\mathrm{x}_{\mathrm{1}} .\mathrm{m}_{\mathrm{1}} \:+\:\mathrm{x}_{\mathrm{2}} .\mathrm{m}_{\mathrm{2}} \:+\:\mathrm{x}_{\mathrm{3}} .\mathrm{m}_{\mathrm{3}} \:+\:....}{\mathrm{m}_{\mathrm{1}} \:+\:\mathrm{m}_{\mathrm{2}} \:+\:\mathrm{m}_{\mathrm{3}} \:+\:......} \\ $$$$\:\:\:\:\:\:\:\mathrm{Y}_{\mathrm{0}} \:=\:\frac{\mathrm{y}_{\mathrm{1}} .\mathrm{m}_{\mathrm{1}} \:+\:\mathrm{y}_{\mathrm{2}} .\mathrm{m}_{\mathrm{2}} \:+\:\mathrm{y}_{\mathrm{3}} .\mathrm{m}_{\mathrm{3}} \:+\:....}{\mathrm{m}_{\mathrm{1}} \:+\:\mathrm{m}_{\mathrm{2}} \:+\:\mathrm{m}_{\mathrm{3}} \:+\:.....} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{X}_{\mathrm{z}} \:=\:\frac{{l}_{\mathrm{1}} .\mathrm{x}_{\mathrm{1}} \:+\:{l}_{\mathrm{2}} \:.\mathrm{x}_{\mathrm{2}} \:+\:{l}_{\mathrm{3}} .\mathrm{x}_{\mathrm{3}} \:+\:.......}{{l}_{\mathrm{1}} \:+\:{l}_{\mathrm{2}} \:+\:{l}_{\mathrm{3}} \:+\:......} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Y}_{\mathrm{z}} \:=\:\frac{{l}_{\mathrm{1}} .{y}_{\mathrm{1}} \:+\:{l}_{\mathrm{2}} .{y}_{\mathrm{2}} \:+\:{l}_{\mathrm{3}} .{y}_{\mathrm{3}} \:+\:......\:}{{l}_{\mathrm{1}} \:+\:{l}_{\mathrm{2}} \:+\:{l}_{\mathrm{3}} \:+\:......} \\ $$$$ \\ $$

Question Number 107905    Answers: 1   Comments: 3

Which of the following set of horizontal forces would lead an object in equilibrium? (a) 5N 10N 20N (b) 6N 12N 18N (c) 8N 8N 8N (b) 2N 4N 8N 16N

$$\mathrm{W}{hich}\:{of}\:{the}\:{following}\:{set}\:{of}\:{horizontal} \\ $$$${forces}\:{would}\:{lead}\:{an}\:{object}\:{in}\:{equilibrium}? \\ $$$$\left({a}\right)\:\mathrm{5}{N}\:\:\:\mathrm{10}{N}\:\:\mathrm{20}{N} \\ $$$$\left({b}\right)\:\mathrm{6}{N}\:\:\:\mathrm{12}{N}\:\:\:\:\mathrm{18}{N} \\ $$$$\left({c}\right)\:\mathrm{8}{N}\:\:\:\mathrm{8}{N}\:\:\:\mathrm{8}{N} \\ $$$$\left({b}\right)\:\mathrm{2}{N}\:\:\:\:\mathrm{4}{N}\:\:\:\mathrm{8}{N}\:\:\:\mathrm{16}{N} \\ $$

Question Number 107900    Answers: 2   Comments: 0

((Σ BeMath Σ)/□) Given tan x−sec x = ϑ then sin x = ?

$$\:\:\:\:\:\:\:\frac{\Sigma\:\mathcal{B}{e}\mathcal{M}{ath}\:\Sigma}{\Box} \\ $$$$\:\:{Given}\:\mathrm{tan}\:{x}−\mathrm{sec}\:{x}\:=\:\vartheta\: \\ $$$$\:{then}\:\mathrm{sin}\:{x}\:=\:? \\ $$

Question Number 107883    Answers: 1   Comments: 0

Expand e^(1/x) (√(x(x+2)))

$$\mathrm{Expand}\:\mathrm{e}^{\mathrm{1}/\mathrm{x}} \sqrt{\mathrm{x}\left(\mathrm{x}+\mathrm{2}\right)} \\ $$

Question Number 107882    Answers: 1   Comments: 5

lim_(x→∞) (((a^(1/x) +b^(1/x) )/2))^x

$$\underset{\mathrm{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{a}^{\mathrm{1}/\mathrm{x}} +\mathrm{b}^{\mathrm{1}/\mathrm{x}} }{\mathrm{2}}\right)^{\mathrm{x}} \\ $$

Question Number 107877    Answers: 0   Comments: 0

Question Number 107867    Answers: 0   Comments: 0

Question Number 107859    Answers: 0   Comments: 0

find A_n =∫_0 ^1 x^n (√(1+x+x^2 ))dx (n natural)

$$\mathrm{find}\:\mathrm{A}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{x}^{\mathrm{n}} \sqrt{\mathrm{1}+\mathrm{x}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\:\:\left(\mathrm{n}\:\mathrm{natural}\right) \\ $$

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