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Question Number 27701 by NECx last updated on 13/Jan/18
If the function f(x) satisfies  lim_(x→1)   ((f(x)−2)/(x^2 −1)) =π, evaluate lim_(x→1) f(x)
$${If}\:{the}\:{function}\:{f}\left({x}\right)\:{satisfies} \\ $$$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\:\frac{{f}\left({x}\right)−\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}}\:=\pi,\:{evaluate}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{f}\left({x}\right) \\ $$
Commented by abdo imad last updated on 21/Jan/18
⇔ lim_(x→1)      ((f(x)−2 −π(x^2 −1))/(x^2 −1))=0 so we must have  lim_(x→1) f(x)−2−π(x^2 −1)=0 in order to find the form (0/0)  ⇒ lim_(x→1) f(x)=2   let verify this number by hospital   theorem lim_(x→1) ((f^′ (x) −2πx)/(2x))=0 ⇒lim_(x→1) f^′ (x)=2π so  we must have lim_(x→1)  f(x)=2 and lim_(x→1) f^′ (x)=2π .
$$\Leftrightarrow\:{lim}_{{x}\rightarrow\mathrm{1}} \:\:\:\:\:\frac{{f}\left({x}\right)−\mathrm{2}\:−\pi\left({x}^{\mathrm{2}} −\mathrm{1}\right)}{{x}^{\mathrm{2}} −\mathrm{1}}=\mathrm{0}\:{so}\:{we}\:{must}\:{have} \\ $$$${lim}_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right)−\mathrm{2}−\pi\left({x}^{\mathrm{2}} −\mathrm{1}\right)=\mathrm{0}\:{in}\:{order}\:{to}\:{find}\:{the}\:{form}\:\frac{\mathrm{0}}{\mathrm{0}} \\ $$$$\Rightarrow\:{lim}_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right)=\mathrm{2}\:\:\:{let}\:{verify}\:{this}\:{number}\:{by}\:{hospital}\: \\ $$$${theorem}\:{lim}_{{x}\rightarrow\mathrm{1}} \frac{{f}^{'} \left({x}\right)\:−\mathrm{2}\pi{x}}{\mathrm{2}{x}}=\mathrm{0}\:\Rightarrow{lim}_{{x}\rightarrow\mathrm{1}} {f}^{'} \left({x}\right)=\mathrm{2}\pi\:{so} \\ $$$${we}\:{must}\:{have}\:{lim}_{{x}\rightarrow\mathrm{1}} \:{f}\left({x}\right)=\mathrm{2}\:{and}\:{lim}_{{x}\rightarrow\mathrm{1}} {f}^{'} \left({x}\right)=\mathrm{2}\pi\:. \\ $$
Answered by mrW2 last updated on 13/Jan/18
lim_(x→1) f(x)=2
$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}{f}\left({x}\right)=\mathrm{2} \\ $$
Commented by NECx last updated on 13/Jan/18
how?
$${how}? \\ $$
Commented by mrW2 last updated on 14/Jan/18
if lim_(x→1)  f(x)≠2,  lim_(x→1)  ((f(x)−2)/(x^2 −1))=((finite)/0)→∞≠π
$${if}\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:{f}\left({x}\right)\neq\mathrm{2}, \\ $$$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)−\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}}=\frac{{finite}}{\mathrm{0}}\rightarrow\infty\neq\pi \\ $$
Commented by abdo imad last updated on 21/Jan/18
this condition is unsufficient....
$${this}\:{condition}\:{is}\:{unsufficient}…. \\ $$

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