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Question Number 78012 by malwaan last updated on 13/Jan/20

prove that  lim_(x→1) ((sin(𝛑cos𝛑x))/((x−1)^2 )) = − (𝛑^3 /2)

$$\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{1}} {\boldsymbol{{lim}}}\frac{\boldsymbol{{sin}}\left(\boldsymbol{\pi{cos}\pi{x}}\right)}{\left(\boldsymbol{{x}}−\mathrm{1}\right)^{\mathrm{2}} }\:=\:−\:\frac{\boldsymbol{\pi}^{\mathrm{3}} }{\mathrm{2}} \\ $$

Commented by msup trace by abdo last updated on 13/Jan/20

let f(x)=((sin(πcos(πx)))/((x−1)^2 ))  chsngement x−1=t give  f(x)=g(t)=((sin(πcos(π(t+1)))/t^2 )  =((sin(−π cos(πt)))/t^2 )  =−((sin(π cos(πt)))/t^2 )  x→1 ⇔t→0 and  cos(πt)∼1−((π^2 t^2 )/2) ⇒  πcos(πt) =π−((π^3 t^2 )/2) ⇒  −πsin(πcos(πt))  ∼−sin(((π^3 t^2 )/2))∼−((π^3 t^2 )/2) ⇒  g(t)∼−(π^3 /2) ⇒lim_(t→0)   g(t)=−(π^3 /2)  lim_x   f(x)=−(π^3 /2)

$${let}\:{f}\left({x}\right)=\frac{{sin}\left(\pi{cos}\left(\pi{x}\right)\right)}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${chsngement}\:{x}−\mathrm{1}={t}\:{give} \\ $$$${f}\left({x}\right)={g}\left({t}\right)=\frac{{sin}\left(\pi{cos}\left(\pi\left({t}+\mathrm{1}\right)\right)\right.}{{t}^{\mathrm{2}} } \\ $$$$=\frac{{sin}\left(−\pi\:{cos}\left(\pi{t}\right)\right)}{{t}^{\mathrm{2}} } \\ $$$$=−\frac{{sin}\left(\pi\:{cos}\left(\pi{t}\right)\right)}{{t}^{\mathrm{2}} } \\ $$$${x}\rightarrow\mathrm{1}\:\Leftrightarrow{t}\rightarrow\mathrm{0}\:{and} \\ $$$${cos}\left(\pi{t}\right)\sim\mathrm{1}−\frac{\pi^{\mathrm{2}} {t}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow \\ $$$$\pi{cos}\left(\pi{t}\right)\:=\pi−\frac{\pi^{\mathrm{3}} {t}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow \\ $$$$−\pi{sin}\left(\pi{cos}\left(\pi{t}\right)\right) \\ $$$$\sim−{sin}\left(\frac{\pi^{\mathrm{3}} {t}^{\mathrm{2}} }{\mathrm{2}}\right)\sim−\frac{\pi^{\mathrm{3}} {t}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow \\ $$$${g}\left({t}\right)\sim−\frac{\pi^{\mathrm{3}} }{\mathrm{2}}\:\Rightarrow{lim}_{{t}\rightarrow\mathrm{0}} \:\:{g}\left({t}\right)=−\frac{\pi^{\mathrm{3}} }{\mathrm{2}} \\ $$$${lim}_{{x}} \:\:{f}\left({x}\right)=−\frac{\pi^{\mathrm{3}} }{\mathrm{2}} \\ $$

Commented by john santu last updated on 13/Jan/20

lim_(x→1)  ((sin (π(−cos (π−πx)))/((x−1)^2 ))  =lim_(y→0)  ((sin (−πcos πy))/y^2 )  let −πcos πy = u ⇒cos πy=((−u)/π)  πy=cos^(−1) (−(u/π)) ⇒y^2 =(1/π^2 )[cos^(−1) (((−u)/π))]^2   lim_(u→−π)  ((π^2  sin (u))/([cos^(−1) (((−u)/π))]^2 )) =(π^2 /(−(2/π)))=−(π^3 /2)

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left(\pi\left(−\mathrm{cos}\:\left(\pi−\pi{x}\right)\right)\right.}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$=\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left(−\pi\mathrm{cos}\:\pi{y}\right)}{{y}^{\mathrm{2}} } \\ $$$${let}\:−\pi\mathrm{cos}\:\pi{y}\:=\:{u}\:\Rightarrow\mathrm{cos}\:\pi{y}=\frac{−{u}}{\pi} \\ $$$$\pi{y}=\mathrm{cos}^{−\mathrm{1}} \left(−\frac{{u}}{\pi}\right)\:\Rightarrow{y}^{\mathrm{2}} =\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\left[\mathrm{cos}^{−\mathrm{1}} \left(\frac{−{u}}{\pi}\right)\right]^{\mathrm{2}} \\ $$$$\underset{{u}\rightarrow−\pi} {\mathrm{lim}}\:\frac{\pi^{\mathrm{2}} \:\mathrm{sin}\:\left({u}\right)}{\left[\mathrm{cos}^{−\mathrm{1}} \left(\frac{−{u}}{\pi}\right)\right]^{\mathrm{2}} }\:=\frac{\pi^{\mathrm{2}} }{−\frac{\mathrm{2}}{\pi}}=−\frac{\pi^{\mathrm{3}} }{\mathrm{2}} \\ $$

Commented by malwaan last updated on 14/Jan/20

thank you sir

$$\boldsymbol{{thank}}\:\boldsymbol{{you}}\:\boldsymbol{{sir}} \\ $$

Answered by MJS last updated on 13/Jan/20

...=lim_(x→1)  (((d^2 /dx^2 )[sin (πcos (πx))])/((d^2 /dx^2 )[(x−1)^2 ]))=  =lim_(x→1)  ((−π^3 cos (πx) ×cos (πcos (πx)) −π^4 sin^2  (πx) ×sin (πcos (πx)))/2)=  =−(π^3 /2)

$$...=\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\frac{{d}^{\mathrm{2}} }{{dx}^{\mathrm{2}} }\left[\mathrm{sin}\:\left(\pi\mathrm{cos}\:\left(\pi{x}\right)\right)\right]}{\frac{{d}^{\mathrm{2}} }{{dx}^{\mathrm{2}} }\left[\left({x}−\mathrm{1}\right)^{\mathrm{2}} \right]}= \\ $$$$=\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{−\pi^{\mathrm{3}} \mathrm{cos}\:\left(\pi{x}\right)\:×\mathrm{cos}\:\left(\pi\mathrm{cos}\:\left(\pi{x}\right)\right)\:−\pi^{\mathrm{4}} \mathrm{sin}^{\mathrm{2}} \:\left(\pi{x}\right)\:×\mathrm{sin}\:\left(\pi\mathrm{cos}\:\left(\pi{x}\right)\right)}{\mathrm{2}}= \\ $$$$=−\frac{\pi^{\mathrm{3}} }{\mathrm{2}} \\ $$

Commented by malwaan last updated on 13/Jan/20

thank you sir MJS  can you solve it without (d/dx)..?

$${thank}\:{you}\:{sir}\:{MJS} \\ $$$${can}\:{you}\:{solve}\:{it}\:{without}\:\frac{{d}}{{dx}}..? \\ $$

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