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Question Number 152211 Answers: 0 Comments: 1

lim_(x→0) (1/x^2 ) should the answer be ∞ or is it DNE. my main question is, when, lim_(x→a) f(x) = ∞ does it not exist? is it DNE?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} } \\ $$$$\mathrm{should}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{be}\:\infty\:\mathrm{or}\:\mathrm{is}\:\mathrm{it}\:\mathrm{DNE}. \\ $$$$\mathrm{my}\:\mathrm{main}\:\mathrm{question}\:\mathrm{is}, \\ $$$$\mathrm{when}, \\ $$$$\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\infty\: \\ $$$$\mathrm{does}\:\mathrm{it}\:\mathrm{not}\:\mathrm{exist}?\:\mathrm{is}\:\mathrm{it}\:\mathrm{DNE}? \\ $$

Question Number 152492 Answers: 1 Comments: 0

A particle is projected upwards with a velocity of 96ms^(−1) . In addition to being subject to gravity, it is acted on by a retardation of 16t, where t is the time from the start of the motion. What is the greatest height attained by the particle?

$$\:\mathrm{A}\:\mathrm{particle}\:\mathrm{is}\:\mathrm{projected}\:\mathrm{upwards}\:\mathrm{with} \\ $$$$\:\mathrm{a}\:\mathrm{velocity}\:\mathrm{of}\:\:\mathrm{96}{ms}^{−\mathrm{1}} .\:\mathrm{In}\:\mathrm{addition}\:\mathrm{to} \\ $$$$\:\mathrm{being}\:\mathrm{subject}\:\mathrm{to}\:\mathrm{gravity},\:\mathrm{it}\:\mathrm{is}\:\mathrm{acted}\:\mathrm{on} \\ $$$$\:\mathrm{by}\:\mathrm{a}\:\mathrm{retardation}\:\mathrm{of}\:\mathrm{16}{t},\:\mathrm{where}\:{t}\:\mathrm{is}\:\mathrm{the} \\ $$$$\:\mathrm{time}\:\mathrm{from}\:\mathrm{the}\:\mathrm{start}\:\mathrm{of}\:\mathrm{the}\:\mathrm{motion}. \\ $$$$\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{height}\:\mathrm{attained} \\ $$$$\:\mathrm{by}\:\mathrm{the}\:\mathrm{particle}? \\ $$

Question Number 152208 Answers: 1 Comments: 0

1.for ∀x>0.find the value of m to 1+log_5 (x^2 +1)≥log_5 (mx^2 +4x+m) verify ∀x.

$$\mathrm{1}.\mathrm{for}\:\forall\mathrm{x}>\mathrm{0}.\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{m}\:\mathrm{to}\: \\ $$$$\mathrm{1}+\mathrm{log}_{\mathrm{5}} \left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\geqslant\mathrm{log}_{\mathrm{5}} \left(\mathrm{mx}^{\mathrm{2}} +\mathrm{4x}+\mathrm{m}\right)\:\mathrm{verify}\:\forall\mathrm{x}. \\ $$

Question Number 152204 Answers: 3 Comments: 2

Question Number 152203 Answers: 0 Comments: 1

If x is real show that (2+i)^((1+3i)x) +(2−i)^((1−3i)x) is also real

$$\mathrm{If}\:\mathrm{x}\:\mathrm{is}\:\mathrm{real}\:\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{2}+\mathrm{i}\right)^{\left(\mathrm{1}+\mathrm{3i}\right)\mathrm{x}} +\left(\mathrm{2}−\mathrm{i}\right)^{\left(\mathrm{1}−\mathrm{3i}\right)\mathrm{x}} \\ $$$$\mathrm{is}\:\mathrm{also}\:\mathrm{real} \\ $$

Question Number 152201 Answers: 0 Comments: 0

...Integral... I := ∫_0 ^( π) ln (sin(x) ).tan^( −1) (cot(x))dx=^? 0 proof :: .... I := ∫_0 ^( π) ln (sin(x) ). tan^( −1) ( tan((π/2) −x ))dx := ∫_0 ^( π) ((π/2) −x ).ln(sin(x))dx := (π/2) ∫_0 ^( π) ln(sin(x))dx−∫_0 ^( π) xln(sin(x))dx := (π/2) (−π ln (2 )) −J ......( 1 ) J : = ∫_0 ^( π) (π − x) ln (sin(x))dx := π (−π ln(2))−J ∴ J :=((−π^( 2) )/2) ln( 2 ) .......(2) (2) ⇛ (1 ) : I = 0 .........■

$$ \\ $$$$\:\:\:...\mathrm{Integral}... \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\pi} {ln}\:\left({sin}\left({x}\right)\:\right).{tan}^{\:−\mathrm{1}} \left({cot}\left({x}\right)\right){dx}\overset{?} {=}\:\mathrm{0} \\ $$$$\:\:\:\:\:{proof}\:::\:.... \\ $$$$\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\pi} {ln}\:\left({sin}\left({x}\right)\:\right).\:{tan}^{\:−\mathrm{1}} \left(\:{tan}\left(\frac{\pi}{\mathrm{2}}\:−{x}\:\right)\right){dx} \\ $$$$\:\:\:\:\:\:\::=\:\int_{\mathrm{0}} ^{\:\pi} \left(\frac{\pi}{\mathrm{2}}\:−{x}\:\right).{ln}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:\:\:\:\::=\:\frac{\pi}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\:\pi} {ln}\left({sin}\left({x}\right)\right){dx}−\int_{\mathrm{0}} ^{\:\pi} {xln}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:\:\:\::=\:\frac{\pi}{\mathrm{2}}\:\left(−\pi\:{ln}\:\left(\mathrm{2}\:\right)\right)\:−\mathrm{J}\:\:\:......\left(\:\mathrm{1}\:\right)\:\: \\ $$$$\:\:\:\:\:\:\mathrm{J}\::\:=\:\int_{\mathrm{0}} ^{\:\pi} \left(\pi\:−\:{x}\right)\:{ln}\:\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:\:\:\:\:\:\::=\:\pi\:\left(−\pi\:{ln}\left(\mathrm{2}\right)\right)−\mathrm{J} \\ $$$$\:\:\:\:\:\:\:\:\therefore\:\:\:\:\:\mathrm{J}\::=\frac{−\pi^{\:\mathrm{2}} }{\mathrm{2}}\:{ln}\left(\:\mathrm{2}\:\right)\:.......\left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\left(\mathrm{2}\right)\:\Rrightarrow\:\left(\mathrm{1}\:\right)\::\:\:\:\:\:\mathrm{I}\:=\:\mathrm{0}\:.........\blacksquare \\ $$$$ \\ $$

Question Number 152178 Answers: 2 Comments: 1

Solve the system { ((y(√x) + x(√y) = x + y)),(((√x) + (√y) = xy)) :} Find all the real solutions other than x = 0 and y = 0

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system} \\ $$$$\begin{cases}{\mathrm{y}\sqrt{\mathrm{x}}\:+\:\mathrm{x}\sqrt{\mathrm{y}}\:=\:\mathrm{x}\:+\:\mathrm{y}}\\{\sqrt{\mathrm{x}}\:+\:\sqrt{\mathrm{y}}\:=\:\mathrm{xy}}\end{cases} \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{real}\:\mathrm{solutions}\:\mathrm{other}\:\mathrm{than} \\ $$$$\mathrm{x}\:=\:\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{y}\:=\:\mathrm{0} \\ $$

Question Number 152175 Answers: 1 Comments: 6

what exact value sin 2x if given cos 3x = (2/( (√5)))

$$\mathrm{what}\:\mathrm{exact}\:\mathrm{value}\:\mathrm{sin}\:\mathrm{2x}\:\mathrm{if}\:\mathrm{given} \\ $$$$\:\mathrm{cos}\:\mathrm{3x}\:=\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{5}}} \\ $$

Question Number 152187 Answers: 1 Comments: 1

Please formular for Γ((8/3))

$$\mathrm{Please}\:\mathrm{formular}\:\mathrm{for}\:\:\:\:\:\Gamma\left(\frac{\mathrm{8}}{\mathrm{3}}\right) \\ $$

Question Number 152165 Answers: 2 Comments: 0

∫((sin x)/(sin 3x))dx