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LimitsQuestion and Answers: Page 1

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Prove; lim_(x→0) ((x − sin x)/x^3 ) = (1/6)

$$ \\ $$$$\:\:\:\:{Prove};\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}\:−\:{sin}\:{x}}{{x}^{\mathrm{3}} }\:=\:\frac{\mathrm{1}}{\mathrm{6}}\:\: \\ $$$$ \\ $$

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lim _(n→∞) (1/n) ( (((2n)!)/(n!)) )^(1/n) = ?

$$ \\ $$$$ \\ $$$$\:\:\:\:\mathrm{lim}\:_{\mathrm{n}\rightarrow\infty} \frac{\mathrm{1}}{{n}}\:\left(\:\frac{\left(\mathrm{2}{n}\right)!}{{n}!}\:\right)^{\frac{\mathrm{1}}{{n}}} =\:?\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 217190 Answers: 1 Comments: 0

Given a_(n+1) = a_n + a_(n+2) where a_3 = 4 and a_5 = 6 find a_n .

$$\mathrm{Given}\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:=\:\mathrm{a}_{\mathrm{n}} \:+\:\mathrm{a}_{\mathrm{n}+\mathrm{2}} \: \\ $$$$\:\:\mathrm{where}\:\mathrm{a}_{\mathrm{3}} =\:\mathrm{4}\:\mathrm{and}\:\mathrm{a}_{\mathrm{5}} =\:\mathrm{6} \\ $$$$\:\mathrm{find}\:\mathrm{a}_{\mathrm{n}} \:. \\ $$

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Evaluate 5^2 Σ_(n=1) ^∞ (1/2)(Σ_(m=2) ^∞ (2/(m^2 +2m)))^(n−1)

$${Evaluate}\:\mathrm{5}^{\mathrm{2}} \underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}}\left(\underset{{m}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{2}}{{m}^{\mathrm{2}} +\mathrm{2}{m}}\right)^{{n}−\mathrm{1}} \\ $$

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lim_(x→+∞) ((√(x+(√(x+(√(x+(√x)))))))−(√x))

$${li}\underset{{x}\rightarrow+\infty} {{m}}\:\left(\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}}}}}−\sqrt{{x}}\right) \\ $$

Question Number 216748 Answers: 1 Comments: 0

Find ∫_(−1) ^1 ln∣Γ((1/2)+it)∣dt

$${Find}\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} {ln}\mid\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}+{it}\right)\mid{dt} \\ $$

Question Number 216274 Answers: 0 Comments: 0

prove that : Σ_(n=1) ^∞ (( cos( n ))/n) ( 1+(1/( (√2))) + (1/( (√3))) + ...+(1/( (√n))) ) is convergent.

$$ \\ $$$$\:\:\:{prove}\:{that}\:: \\ $$$$ \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{cos}\left(\:{n}\:\right)}{{n}}\:\left(\:\mathrm{1}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:+\:...+\frac{\mathrm{1}}{\:\sqrt{{n}}}\:\right) \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:{is}\:\:\:{convergent}. \\ $$$$ \\ $$

Question Number 216055 Answers: 1 Comments: 0

lim_(x→0) ((sin^2 2x)/( ((cos x))^(1/3) −((cos x))^(1/4) )) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2x}}{\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\mathrm{x}}−\sqrt[{\mathrm{4}}]{\mathrm{cos}\:\mathrm{x}}}\:=? \\ $$

Question Number 216032 Answers: 2 Comments: 0

lim_(x→0) ((1−cos x (√(cos 2x)))/x^2 ) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}\:\sqrt{\mathrm{cos}\:\mathrm{2x}}}{\mathrm{x}^{\mathrm{2}} }\:=? \\ $$

Question Number 216014 Answers: 1 Comments: 0

lim_(Δx→cos(π/2)) ((sin^3 (Δx+x)−sin^3 x)/(2^(−1) ∙Δx))=?

$$\underset{\Delta{x}\rightarrow{cos}\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\mathrm{sin}^{\mathrm{3}} \left(\Delta{x}+{x}\right)−{sin}^{\mathrm{3}} {x}}{\mathrm{2}^{−\mathrm{1}} \centerdot\Delta{x}}=? \\ $$

Question Number 215884 Answers: 1 Comments: 0

lim_(x→0) ((cos 2x−cos 6x)/(1−cos 3x cos 5x)) =?

$$\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{2x}−\mathrm{cos}\:\mathrm{6x}}{\mathrm{1}−\mathrm{cos}\:\mathrm{3x}\:\mathrm{cos}\:\mathrm{5x}}\:=? \\ $$$$\:\:\:\: \\ $$

Question Number 215831 Answers: 2 Comments: 0

lim_(x→∞) (((√((x+1)^3 ))−(√((x−1)^3 )))/( (√x))) =?

$$\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} }−\sqrt{\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{3}} }}{\:\sqrt{\mathrm{x}}}\:=? \\ $$

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lim_(n→∞) [Π_(k=1) ^(n+1) Γ((1/k))]^(1/(n+1)) −[Π_(k=1) ^n Γ((1/k))]^(1/n)

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left[\underset{{k}=\mathrm{1}} {\overset{{n}+\mathrm{1}} {\prod}}\Gamma\left(\frac{\mathrm{1}}{{k}}\right)\right]^{\frac{\mathrm{1}}{{n}+\mathrm{1}}} −\left[\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\Gamma\left(\frac{\mathrm{1}}{{k}}\right)\right]^{\frac{\mathrm{1}}{{n}}} \\ $$

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lim_(x→∞) (ln(9/2))^((1/2)x) =?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left({ln}\frac{\mathrm{9}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}{x}} =? \\ $$$$ \\ $$

Question Number 215667 Answers: 2 Comments: 0

lim_(x→1) sec((π/2)x)(arctanx−(π/4))=?

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{sec}\left(\frac{\pi}{\mathrm{2}}{x}\right)\left({arctanx}−\frac{\pi}{\mathrm{4}}\right)=? \\ $$

Question Number 215343 Answers: 1 Comments: 1

lim_(x→0) ((sin 3xcos5x )/x^2 )

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:\mathrm{3}{x}\mathrm{cos5}{x}\:}{{x}^{\mathrm{2}} } \\ $$

Question Number 215808 Answers: 1 Comments: 0

⇃

$$\:\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 215051 Answers: 1 Comments: 0

lim_(n→∞) Π_(k=1) ^n ((k^2 +1)/( (√(k^2 +4))))=?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\frac{{k}^{\mathrm{2}} +\mathrm{1}}{\:\sqrt{{k}^{\mathrm{2}} +\mathrm{4}}}=? \\ $$$$ \\ $$

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