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LimitsQuestion and Answers: Page 1

Question Number 224288 Answers: 1 Comments: 2

P= Π_(k=1) ^∞ (1/( (√(1+(1/k))) (1−(1/(2k))))) =?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{P}=\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\prod}}\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\frac{\mathrm{1}}{{k}}}\:\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}{k}}\right)}\:=?\:\:\:\: \\ $$$$ \\ $$

Question Number 223990 Answers: 5 Comments: 1

lim_(x→1) ((x(x+(1/x))^5 −32 )/(x−1))

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{{x}\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{5}} −\mathrm{32}\:}{{x}−\mathrm{1}} \\ $$

Question Number 223193 Answers: 0 Comments: 0

Prove that: ∫_( 0) ^( (π/2)) tan^(− 1) (r sin θ) dθ = 2𝛘_2 ((((√(1 + r^2 )) − 1)/r))

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\:\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\mathrm{tan}^{−\:\mathrm{1}} \left(\mathrm{r}\:\mathrm{sin}\:\theta\right)\:\mathrm{d}\theta\:\:\:=\:\:\:\mathrm{2}\boldsymbol{\chi}_{\mathrm{2}} \left(\frac{\sqrt{\mathrm{1}\:\:+\:\:\mathrm{r}^{\mathrm{2}} }\:\:−\:\:\mathrm{1}}{\mathrm{r}}\right) \\ $$

Question Number 223082 Answers: 0 Comments: 0

if lim_(x→+∞) x−f(x)=+∞ and lim_(x→+∞) x+f(x)=+∞ can we determine lim_(x→+∞) ((x−f(x))/(x+f(x)))

$${if}\:\underset{{x}\rightarrow+\infty} {{lim}x}−{f}\left({x}\right)=+\infty\:{and}\:\underset{{x}\rightarrow+\infty} {{lim}x}+{f}\left({x}\right)=+\infty \\ $$$${can}\:{we}\:{determine}\:\underset{{x}\rightarrow+\infty} {{lim}}\frac{{x}−{f}\left({x}\right)}{{x}+{f}\left({x}\right)} \\ $$$$ \\ $$

Question Number 222756 Answers: 1 Comments: 0

lim_(x→0) ((1−(√(cos(x))))/( x−xcos((√x))))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\sqrt{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}}{\:\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{xcos}}\left(\sqrt{\boldsymbol{\mathrm{x}}}\right)} \\ $$

Question Number 222478 Answers: 1 Comments: 0

S=Σ_(n=1) ^∞ (−1)^(n−1) (H_n /n^2 ) = ? note: H_n =1+(1/2) +(1/3) +...+(1/n)

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{S}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \frac{{H}_{{n}} }{{n}^{\mathrm{2}} }\:=\:? \\ $$$$\:{note}:\:\:\:{H}_{{n}} =\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{3}}\:+...+\frac{\mathrm{1}}{{n}}\: \\ $$

Question Number 222261 Answers: 1 Comments: 0

lim_(x→0) ((tan(x^2 +4x))/(sin(9x^2 +x))) No L′ho^ pital′s rule allowed!

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{tan}\left({x}^{\mathrm{2}} +\mathrm{4}{x}\right)}{\mathrm{sin}\left(\mathrm{9}{x}^{\mathrm{2}} +{x}\right)} \\ $$$$\mathrm{No}\:\mathrm{L}'\mathrm{h}\hat {\mathrm{o}pital}'\mathrm{s}\:\mathrm{rule}\:\mathrm{allowed}! \\ $$

Question Number 221392 Answers: 1 Comments: 0

lim_(x→3) (√(x−3))=? 1) 0 2) 3 3) Does not exist 4) Undefined

$$\underset{{x}\rightarrow\mathrm{3}} {\mathrm{lim}}\sqrt{{x}−\mathrm{3}}=? \\ $$$$\left.\mathrm{1}\right)\:\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{3} \\ $$$$\left.\mathrm{3}\right)\:{Does}\:{not}\:{exist} \\ $$$$\left.\mathrm{4}\right)\:{Undefined} \\ $$

Question Number 221348 Answers: 2 Comments: 0

lim_(x→2) ((4−2^x )/(x−2))

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{4}−\mathrm{2}^{{x}} }{{x}−\mathrm{2}} \\ $$

Question Number 221347 Answers: 1 Comments: 0

lim_(x→2) ((4−x^2 )/(x−2))

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{4}−{x}^{\mathrm{2}} }{{x}−\mathrm{2}} \\ $$

Question Number 221260 Answers: 1 Comments: 0

Question Number 221047 Answers: 1 Comments: 0

Question Number 221034 Answers: 1 Comments: 0

Question Number 220852 Answers: 1 Comments: 0

Lim_(x→0) {((xe^x −log(1+x))/x^2 )}

$$\underset{{x}\rightarrow\mathrm{0}} {{Lim}}\left\{\frac{{xe}^{{x}} −{log}\left(\mathrm{1}+{x}\right)}{{x}^{\mathrm{2}} }\right\} \\ $$

Question Number 220843 Answers: 0 Comments: 1

α ∈ R lim_(x→1) (((1 − x)^α )/(^3 (√(1 − x^4 )))) ∈(0,∞)

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\alpha\:\in\:\mathbb{R} \\ $$$$\:\:\:\:\:\mathrm{lim}_{{x}\rightarrow\mathrm{1}} \:\frac{\left(\mathrm{1}\:−\:{x}\right)^{\alpha} }{\:^{\mathrm{3}} \sqrt{\mathrm{1}\:−\:{x}^{\mathrm{4}} }}\:\:\:\:\:\:\:\:\in\left(\mathrm{0},\infty\right) \\ $$$$ \\ $$

Question Number 220811 Answers: 0 Comments: 0

Question Number 220764 Answers: 1 Comments: 0

L= lim _( n→∞) (Σ_(k=1) ^n (k/(n^2 +k^2 ))).(∫^( 1) _( 0) e^(−x^2 ) dx)^(−1) .(Σ_(m=0) ^∞ (((−1)^m )/((2m+1)3^m )))

$$ \\ $$$$\:\:\boldsymbol{\mathrm{L}}=\:\boldsymbol{\mathrm{lim}}\underset{\:\boldsymbol{{n}}\rightarrow\infty} {\:}\left(\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\:\frac{\boldsymbol{{k}}}{\boldsymbol{{n}}^{\mathrm{2}} +\boldsymbol{{k}}^{\mathrm{2}} }\right).\left(\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \boldsymbol{{e}}^{−\boldsymbol{{x}}^{\mathrm{2}} } \boldsymbol{{dx}}\overset{−\mathrm{1}} {\right)}.\left(\underset{\boldsymbol{{m}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{\boldsymbol{{m}}} }{\left(\mathrm{2}\boldsymbol{{m}}+\mathrm{1}\right)\mathrm{3}^{\boldsymbol{{m}}} }\right)\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220380 Answers: 3 Comments: 0

lim_(n→∞) tan[(π/4)+(1/n)]^n =?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}tan}\left[\frac{\pi}{\mathrm{4}}+\frac{\mathrm{1}}{{n}}\right]^{{n}} =? \\ $$

Question Number 219832 Answers: 1 Comments: 0

lim_(n→∞) n((1/(1+n)) +(1/(2+n)) +...+(1/(2n)) −ln(2))=?

$$ \\ $$$$\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \:{n}\left(\frac{\mathrm{1}}{\mathrm{1}+{n}}\:+\frac{\mathrm{1}}{\mathrm{2}+{n}}\:+...+\frac{\mathrm{1}}{\mathrm{2}{n}}\:−{ln}\left(\mathrm{2}\right)\right)=? \\ $$$$ \\ $$

Question Number 219731 Answers: 1 Comments: 0

Question Number 219365 Answers: 0 Comments: 0

Question Number 219223 Answers: 2 Comments: 0

Question Number 218703 Answers: 3 Comments: 0

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

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

Question Number 218543 Answers: 1 Comments: 1

Question Number 218196 Answers: 3 Comments: 0

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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