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LimitsQuestion and Answers: Page 1

Question Number 208482    Answers: 0   Comments: 0

Question Number 208093    Answers: 1   Comments: 0

lim_(n→∞) (1/n)((a+(1/n))^2 +(a+(2/n))^2 +...+(a+((n−1)/n))^2 )

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{{n}}\left(\left({a}+\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} +\left({a}+\frac{\mathrm{2}}{{n}}\right)^{\mathrm{2}} +...+\left({a}+\frac{{n}−\mathrm{1}}{{n}}\right)^{\mathrm{2}} \right) \\ $$

Question Number 207816    Answers: 1   Comments: 1

lim_(x→2) ((((x^2 +4))^(1/3) −(√(x^3 −4)))/( (√(x^2 −4))−((x−2))^(1/3) ))

$$\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{x}^{\mathrm{2}} +\mathrm{4}}−\sqrt{\mathrm{x}^{\mathrm{3}} −\mathrm{4}}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{4}}−\sqrt[{\mathrm{3}}]{\mathrm{x}−\mathrm{2}}} \\ $$

Question Number 207528    Answers: 2   Comments: 0

Question Number 207339    Answers: 2   Comments: 0

lim_(x→∞) (((x+a)^(1/x) −x^(1/x) )/((x+b)^(1/x) −x^(1/x) )) =?

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

Question Number 206992    Answers: 1   Comments: 0

$$ \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 206934    Answers: 0   Comments: 3

Question Number 206730    Answers: 1   Comments: 1

find lim_(n→+∞) ∫_0 ^n e^(nx) arctan((x/n))dx

$${find}\:{lim}_{{n}\rightarrow+\infty} \int_{\mathrm{0}} ^{{n}} {e}^{{nx}} \:{arctan}\left(\frac{{x}}{{n}}\right){dx} \\ $$

Question Number 206702    Answers: 2   Comments: 0

lim_(n→∞) (√(cosn+sinn−3^n +4^n ))

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{cos}{n}+\mathrm{sin}{n}−\mathrm{3}^{{n}} +\mathrm{4}^{{n}} } \\ $$

Question Number 206433    Answers: 2   Comments: 0

let f:[0,∞)→R be a continuous function if lim_(n→∞ ) ∫_0 ^1 f(x+n)dx = 2 then lim_(n→∞) f(nx) = ?

$$\:\:\:\:\:\mathrm{let}\:\mathrm{f}:\left[\mathrm{0},\infty\right)\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{continuous}\:\mathrm{function}\:\mathrm{if} \\ $$$$\:\:\:\:\underset{\mathrm{n}\rightarrow\infty\:} {\mathrm{lim}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{f}\left(\mathrm{x}+\mathrm{n}\right)\mathrm{dx}\:=\:\mathrm{2} \\ $$$$\:\mathrm{then}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{nx}\right)\:=\:? \\ $$$$\: \\ $$

Question Number 206095    Answers: 2   Comments: 0

Question Number 206069    Answers: 1   Comments: 0

lim_(x→0) ((−x^3 +x)/(sin x))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−{x}^{\mathrm{3}} +{x}}{\mathrm{sin}\:{x}} \\ $$

Question Number 205916    Answers: 1   Comments: 0

lim_(x→0^+ ) xln(e^x −1)

$${lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:{xln}\left({e}^{{x}} −\mathrm{1}\right) \\ $$

Question Number 205774    Answers: 2   Comments: 0

−−−−−−− Ω = Σ_(n=0) ^∞ (( (−1)^( n) )/((−1)^( n) −n)) = ? −−−−−−−

$$ \\ $$$$\:\:\:\:\:\:−−−−−−− \\ $$$$\:\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\:\left(−\mathrm{1}\right)^{\:{n}} }{\left(−\mathrm{1}\right)^{\:{n}} \:−{n}}\:=\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:−−−−−−− \\ $$

Question Number 205727    Answers: 1   Comments: 4

calculate lim_(x→0) ((e^x −cosx)/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{e}^{{x}} −{cosx}}{{x}^{\mathrm{2}} } \\ $$

Question Number 205716    Answers: 2   Comments: 0

lim_(n→∞) (((2n+1)(2n+3)...(4n+1))/((2n)(2n+2)...(4n))) = ?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{3}\right)...\left(\mathrm{4}{n}+\mathrm{1}\right)}{\left(\mathrm{2}{n}\right)\left(\mathrm{2}{n}+\mathrm{2}\right)...\left(\mathrm{4}{n}\right)}\:\:=\:\:? \\ $$

Question Number 205580    Answers: 1   Comments: 2

lim_(n→∞) (((2n+1)(2n+3)...(4n+1))/((2n)(2n+2)...(4n))) = ?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{3}\right)...\left(\mathrm{4}{n}+\mathrm{1}\right)}{\left(\mathrm{2}{n}\right)\left(\mathrm{2}{n}+\mathrm{2}\right)...\left(\mathrm{4}{n}\right)}\:\:=\:\:? \\ $$

Question Number 205448    Answers: 0   Comments: 0

A=lim_(x→0) ((sinx)/x^3 )=?

$${A}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{sinx}}{{x}^{\mathrm{3}} }=? \\ $$

Question Number 205379    Answers: 2   Comments: 0

Question Number 205339    Answers: 1   Comments: 0

lim_(x→0) ((tan(tanx))/(sin(1−cosx)))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{tan}\left(\mathrm{tan}{x}\right)}{\mathrm{sin}\left(\mathrm{1}−\mathrm{cos}{x}\right)} \\ $$

Question Number 205307    Answers: 1   Comments: 1

lim_(n→∞) ((⌊a⌋+⌊2a⌋+...+⌊na⌋)/n^2 ) where a∈R and ⌊x⌋ is the floor of x ∈ R

$$\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\lfloor{a}\rfloor+\lfloor\mathrm{2}{a}\rfloor+...+\lfloor{na}\rfloor}{{n}^{\mathrm{2}} }\:\mathrm{where}\:{a}\in\mathbb{R} \\ $$$$\:\:\:\mathrm{and}\:\lfloor{x}\rfloor\:\mathrm{is}\:\mathrm{the}\:\mathrm{floor}\:\mathrm{of}\:\mathrm{x}\:\in\:\mathbb{R} \\ $$

Question Number 205237    Answers: 1   Comments: 0

Question Number 205142    Answers: 1   Comments: 0

lim_(n→∞) n^(−n^2 ) [(n+1)(n+(1/2))(n+(1/2^2 ))...(n+(1/2^(n−1) ))]^n =?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}^{−\mathrm{n}^{\mathrm{2}} } \left[\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\right)...\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}−\mathrm{1}} }\right)\right]^{\mathrm{n}} =? \\ $$

Question Number 205134    Answers: 1   Comments: 0

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 204991    Answers: 2   Comments: 1

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