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Question Number 153765 Answers: 2 Comments: 0

Given f:R→R is increasing positive function with lim_(x→∞) ((f(3x))/(f(x)))=1 . What the value of lim_(x→∞) ((f(2x))/(f(x))). (A) 3 (B) (3/2) (C) 1 (D)(2/3) (E) ∞

$$\:{Given}\:{f}:{R}\rightarrow{R}\:{is}\:{increasing}\:{positive} \\ $$$${function}\:{with}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left(\mathrm{3}{x}\right)}{{f}\left({x}\right)}=\mathrm{1}\:.\: \\ $$$${What}\:{the}\:{value}\:{of}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left(\mathrm{2}{x}\right)}{{f}\left({x}\right)}. \\ $$$$\left({A}\right)\:\mathrm{3}\:\:\:\:\:\left({B}\right)\:\frac{\mathrm{3}}{\mathrm{2}}\:\:\:\:\:\left({C}\right)\:\mathrm{1}\:\:\:\:\:\left({D}\right)\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\left({E}\right)\:\infty \\ $$

Question Number 153764 Answers: 1 Comments: 2

Prove without any software: ∫_( 0) ^( 1) ∫_( 0) ^( 1) (√(1 - (((x + y)/2))^2 )) dxdy > (π/4)

$$\mathrm{Prove}\:\mathrm{without}\:\mathrm{any}\:\mathrm{software}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\sqrt{\mathrm{1}\:-\:\left(\frac{\mathrm{x}\:+\:\mathrm{y}}{\mathrm{2}}\right)^{\mathrm{2}} }\:\mathrm{dxdy}\:>\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 153763 Answers: 1 Comments: 0

Determine all pairs (x;y) of integers which satisfy ∣x^2 - y^2 ∣ - (√(16y + 1)) = 0

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{pairs}\:\left(\mathrm{x};\mathrm{y}\right)\:\mathrm{of}\:\mathrm{integers} \\ $$$$\mathrm{which}\:\mathrm{satisfy} \\ $$$$\mid\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{y}^{\mathrm{2}} \mid\:-\:\sqrt{\mathrm{16y}\:+\:\mathrm{1}}\:=\:\mathrm{0} \\ $$

Question Number 153760 Answers: 1 Comments: 0

∫ sin^2 4x cos 4x dx=

$$\int\:\mathrm{sin}^{\mathrm{2}} \mathrm{4}{x}\:\mathrm{cos}\:\mathrm{4}{x}\:{dx}= \\ $$

Question Number 153759 Answers: 0 Comments: 0

Ω= Σ_(n=1) ^∞ {n^2 (∫_0 ^( (π/2)) (( sin^( 2) (x ))/((sin(x)+cos(x))^( 4) )))^( n) dx}=?

$$ \\ $$$$\Omega=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left\{{n}^{\mathrm{2}} \left(\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{sin}^{\:\mathrm{2}} \left({x}\:\right)}{\left({sin}\left({x}\right)+{cos}\left({x}\right)\right)^{\:\mathrm{4}} }\right)^{\:{n}} {dx}\right\}=? \\ $$$$ \\ $$

Question Number 153757 Answers: 1 Comments: 1

Question Number 153755 Answers: 0 Comments: 0

Question Number 153742 Answers: 1 Comments: 0

Question Number 153737 Answers: 2 Comments: 0

Show that ∫_0 ^( (π/2)) (1/((cos θ+ (√3) sin θ)^2 )) dθ= (1/( (√3) ))

$$\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}}{\left(\mathrm{cos}\:\theta+\:\sqrt{\mathrm{3}}\:\mathrm{sin}\:\theta\right)^{\mathrm{2}} }\:{d}\theta=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}\:} \\ $$

Question Number 153734 Answers: 0 Comments: 0

Question Number 153733 Answers: 0 Comments: 0

Question Number 153728 Answers: 0 Comments: 0

Question Number 153722 Answers: 0 Comments: 0

lim_(x→0) ((tan x+x sec x−sin x−x)/(x^3 cos x)) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:{x}+{x}\:\mathrm{sec}\:{x}−\mathrm{sin}\:{x}−{x}}{{x}^{\mathrm{3}} \:\mathrm{cos}\:{x}}\:=? \\ $$

Question Number 153721 Answers: 2 Comments: 0

prove that : I:= ∫_0 ^( ∞) (( x^( 3) )/(sinh ( x ))) dx = ((π^4 )/8) ■ m.n

$$ \\ $$$$\:\:\:\:\:\mathrm{prove}\:\:\mathrm{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{x}^{\:\mathrm{3}} }{{sinh}\:\left(\:{x}\:\right)}\:{dx}\:=\:\frac{\pi\:^{\mathrm{4}} }{\mathrm{8}}\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 153716 Answers: 1 Comments: 1

Question Number 153714 Answers: 2 Comments: 0

How many three−digit numbers can be formed using the digits 2,3,5,7,8 if the number is odd and no digit is repeted?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{three}−\mathrm{digit}\:\mathrm{numbers}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{formed}\:\mathrm{using}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{7},\mathrm{8}\:\mathrm{if}\: \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{and}\:\mathrm{no}\:\mathrm{digit}\:\mathrm{is}\:\mathrm{repeted}? \\ $$

Question Number 153708 Answers: 2 Comments: 0

Question Number 153704 Answers: 1 Comments: 1

Question Number 153696 Answers: 1 Comments: 0

Question Number 153867 Answers: 0 Comments: 2

Question Number 153698 Answers: 2 Comments: 0

solve in R { ((x^3 −y^3 =19)),((xy=6)) :}

$${solve}\:{in}\:\mathbb{R} \\ $$$$\begin{cases}{{x}^{\mathrm{3}} −{y}^{\mathrm{3}} =\mathrm{19}}\\{{xy}=\mathrm{6}}\end{cases} \\ $$

Question Number 153685 Answers: 3 Comments: 2

Question Number 153682 Answers: 1 Comments: 0

((i)^(1/i) )^(xi) = i^x x=?

$$\:\left(\sqrt[{{i}}]{{i}}\:\right)^{{xi}} \:=\:{i}^{{x}} \: \\ $$$$\:\:{x}=?\: \\ $$

Question Number 153681 Answers: 1 Comments: 0

24^(log _(10) (x)) −26^(log _(10) (x)) =1 x=?

$$\:\:\mathrm{24}^{\mathrm{log}\:_{\mathrm{10}} \left({x}\right)} −\mathrm{26}^{\mathrm{log}\:_{\mathrm{10}} \left({x}\right)} =\mathrm{1} \\ $$$$\:{x}=? \\ $$

Question Number 153679 Answers: 1 Comments: 0

Find the constant of polynom P(11x−2) if given the equation 3P(x+2)−P(2x+3)=−4x^2 −x+3

$${Find}\:{the}\:{constant}\:{of}\:{polynom} \\ $$$$\:{P}\left(\mathrm{11}{x}−\mathrm{2}\right)\:{if}\:{given}\:{the}\:{equation} \\ $$$$\mathrm{3}{P}\left({x}+\mathrm{2}\right)−{P}\left(\mathrm{2}{x}+\mathrm{3}\right)=−\mathrm{4}{x}^{\mathrm{2}} −{x}+\mathrm{3} \\ $$

Question Number 153676 Answers: 2 Comments: 0

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