Question 1199731
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Find: limit   {{{((2pi)/x) * (1/(sin((pi*x)/(x - 1)))))}}}   when  x→ + ∞ 
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<pre>
{{{(pi*x)/(x-1)}}} = {{{(pi*(x-1))/(x-1)}}} + {{{pi/(x-1)}}} = {{{pi}}} + {{{pi/(x-1)}}}.    (1)


Therefore,  sin((pi*x)/(x-1)) = sin(pi + pi/(x-1)) = {{{-sin(pi/(x-1))}}}.    (2)


When x→ + ∞,  (x-1) is large value;  {{{pi/(x-1)}}} is a small value;

therefore, {{{sin(pi/(x-1))}}} is a small value equivalent to {{{pi/(x-1)}}}. 


It implies that  {{{1/sin((pi*x/(x - 1)))}}}  is a negative value equivalent to  {{{-(x-1)/pi}}}.


Then  {{{(2pi/x) * (1/sin((pi*x)/(x - 1)))}}}  is equivalent to  {{{(2pi/x)*(-(x-1)/pi)}}}.


As x→ + ∞,  the quantity  {{{(2pi/x)*(-(x-1)/pi)}}}  tends to -2.


Thus limit {{{(2pi/x) * (1/(sin((pi*x/(x - 1)))))}}}  is  -2  when  (x→ + ∞).
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Solved.