Question and Answers Forum

A=lim_(x→0 ) ((1−cos2x)/(2x^2 )) =lim_(x→0) ((2sin^2 x)/(2x^2 )) =lim_(x→0) (((sinx)/x))^2 =1 B=lim_(x→0) (1/(xcotx)) =lim_(x→0) ((tanx)/x)=lim_(x→0) ((sinx)/x)×(1/(cosx))=1

$${A}=\underset{{x}\rightarrow\mathrm{0}\:\:} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos2}{x}}{\mathrm{2}{x}^{\mathrm{2}} } \\ $$$$\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2sin}^{\mathrm{2}} {x}}{\mathrm{2}{x}^{\mathrm{2}} } \\ $$$$\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{sin}{x}}{{x}}\right)^{\mathrm{2}} =\mathrm{1} \\ $$$${B}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{{x}\mathrm{cot}{x}} \\ $$$$\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}{x}}{{x}}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}{x}}{{x}}×\frac{\mathrm{1}}{\mathrm{cos}{x}}=\mathrm{1} \\ $$

Question Number 205135 Answers: 1 Comments: 1

Question Number 205134 Answers: 1 Comments: 0

Question Number 205130 Answers: 1 Comments: 1

Question Number 205117 Answers: 1 Comments: 0

Question Number 205116 Answers: 2 Comments: 0

let x^2 −3x+p = 0 has two positive roots ′a′ and ′b′ then inf((4/a)+(1/b)) is

$$\:\:\mathrm{let}\:\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{p}\:=\:\mathrm{0}\:\mathrm{has}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{roots} \\ $$$$\:'\mathrm{a}'\:\mathrm{and}\:'\mathrm{b}'\:\mathrm{then}\:\:\mathrm{inf}\left(\frac{\mathrm{4}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}\right)\:\mathrm{is}\: \\ $$

Question Number 205114 Answers: 1 Comments: 0

Solve: lim_((x,y)→(0,0)) ((1−cos((√(10xy))))/(3.y.sin(22x))) Ans.: (5/(66)) Step by step, please!

$${Solve}: \\ $$$$ \\ $$$$\:\:{lim}_{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} \frac{\mathrm{1}−{cos}\left(\sqrt{\mathrm{10}{xy}}\right)}{\mathrm{3}.{y}.{sin}\left(\mathrm{22}{x}\right)} \\ $$$$ \\ $$$${Ans}.:\:\frac{\mathrm{5}}{\mathrm{66}} \\ $$$${Step}\:{by}\:{step},\:{please}! \\ $$

Question Number 205107 Answers: 0 Comments: 2

y = log_2 (sin(x)+cos(x)) ⇒ R_y = ?(Range )

$$ \\ $$$$\:\:\:\:{y}\:=\:{log}_{\mathrm{2}} \left({sin}\left({x}\right)+{cos}\left({x}\right)\right) \\ $$$$\:\:\:\Rightarrow\:\:{R}_{{y}} \:=\:?\left({Range}\:\right) \\ $$$$ \\ $$

Question Number 205106 Answers: 1 Comments: 1

Question Number 205101 Answers: 1 Comments: 0

given that there are real constant a,b, c, d such the identity λx^2 +2xy+y^2 = (ax+by)^2 +(cx+dy)^2 holds for all x,y ∈ R this implies (a) λ=−5 (b) λ≥1 (c)0<λ<1 (d) there is no such λ∈R

$$\:\:\mathrm{given}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{real}\:\mathrm{constant}\:\mathrm{a},\mathrm{b},\:\mathrm{c},\:\mathrm{d} \\ $$$$\:\:\mathrm{such}\:\mathrm{the}\:\mathrm{identity} \\ $$$$\:\lambda\mathrm{x}^{\mathrm{2}} +\mathrm{2xy}+\mathrm{y}^{\mathrm{2}} =\:\left(\mathrm{ax}+\mathrm{by}\right)^{\mathrm{2}} +\left(\mathrm{cx}+\mathrm{dy}\right)^{\mathrm{2}} \:\mathrm{holds} \\ $$$$\:\mathrm{for}\:\mathrm{all}\:\mathrm{x},\mathrm{y}\:\in\:\mathbb{R}\:\mathrm{this}\:\mathrm{implies} \\ $$$$\left({a}\right)\:\lambda=−\mathrm{5}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({b}\right)\:\lambda\geqslant\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\left({c}\right)\mathrm{0}<\lambda<\mathrm{1} \\ $$$$\:\left({d}\right)\:\mathrm{there}\:\mathrm{is}\:\mathrm{no}\:\mathrm{such}\:\lambda\in\mathbb{R} \\ $$

Question Number 205091 Answers: 0 Comments: 0

f:z ⇒ z f:z ⇒ z_n f:z_n ⇒ z_n How many homomorphism can be define

$${f}:{z}\:\Rightarrow\:{z} \\ $$$${f}:{z}\:\Rightarrow\:{z}_{{n}} \\ $$$${f}:{z}_{{n}} \Rightarrow\:{z}_{{n}} \\ $$$${How}\:{many}\:{homomorphism}\:{can}\:{be}\:{define} \\ $$

Question Number 205083 Answers: 1 Comments: 0

Question Number 205073 Answers: 6 Comments: 0

if a, b, c are the roots of f(x)=x^3 −2024x^2 +2024x+2024 find (1/(1−a^2 ))+(1/(1−b^2 ))+(1/(1−c^2 ))=?

$${if}\:{a},\:{b},\:{c}\:{are}\:{the}\:{roots}\:{of} \\ $$$${f}\left({x}\right)={x}^{\mathrm{3}} −\mathrm{2024}{x}^{\mathrm{2}} +\mathrm{2024}{x}+\mathrm{2024} \\ $$$${find}\:\frac{\mathrm{1}}{\mathrm{1}−{a}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{1}−{b}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{1}−{c}^{\mathrm{2}} }=? \\ $$

Question Number 205070 Answers: 0 Comments: 5

Given { ((A∩B= { a, b})),((A∩C = { b, c} )),((B∩C= { b ,d })) :} then (A∩C) + (A∩B) + (B∩C)

$$\:\:\:\mathrm{Given}\:\begin{cases}{\mathrm{A}\cap\mathrm{B}=\:\left\{\:\mathrm{a},\:\mathrm{b}\right\}}\\{\mathrm{A}\cap\mathrm{C}\:=\:\left\{\:\mathrm{b},\:\mathrm{c}\right\}\:}\\{\mathrm{B}\cap\mathrm{C}=\:\left\{\:\mathrm{b}\:,\mathrm{d}\:\right\}}\end{cases} \\ $$$$\:\:\:\:\mathrm{then}\:\left(\mathrm{A}\cap\mathrm{C}\right)\:+\:\left(\mathrm{A}\cap\mathrm{B}\right)\:+\:\left(\mathrm{B}\cap\mathrm{C}\right) \\ $$

Question Number 205092 Answers: 2 Comments: 0

Question Number 205062 Answers: 3 Comments: 0

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