SOLUTION: Find: limt((2π/x) * (1/(sin((πx/(x - 1)))))), when (x→ + ∞)

Algebra ->  Customizable Word Problem Solvers  -> Finance -> SOLUTION: Find: limt((2π/x) * (1/(sin((πx/(x - 1)))))), when (x→ + ∞)       Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 1199731: Find:
limt((2π/x) * (1/(sin((πx/(x - 1)))))), when (x→ + ∞)
Answer by ikleyn(52767) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find: limit   %28%282pi%29%2Fx%29+%2A+%281%2F%28sin%28%28pi%2Ax%29%2F%28x+-+1%29%29%29%29%29   when  x→ + ∞
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

%28pi%2Ax%29%2F%28x-1%29 = %28pi%2A%28x-1%29%29%2F%28x-1%29 + pi%2F%28x-1%29 = pi + pi%2F%28x-1%29.    (1)


Therefore,  sin((pi*x)/(x-1)) = sin(pi + pi/(x-1)) = -sin%28pi%2F%28x-1%29%29.    (2)


When x→ + ∞,  (x-1) is large value;  pi%2F%28x-1%29 is a small value;

therefore, sin%28pi%2F%28x-1%29%29 is a small value equivalent to pi%2F%28x-1%29. 


It implies that  1%2Fsin%28%28pi%2Ax%2F%28x+-+1%29%29%29  is a negative value equivalent to  -%28x-1%29%2Fpi.


Then  %282pi%2Fx%29+%2A+%281%2Fsin%28%28pi%2Ax%29%2F%28x+-+1%29%29%29  is equivalent to  %282pi%2Fx%29%2A%28-%28x-1%29%2Fpi%29.


As x→ + ∞,  the quantity  %282pi%2Fx%29%2A%28-%28x-1%29%2Fpi%29  tends to -2.


Thus limit %282pi%2Fx%29+%2A+%281%2F%28sin%28%28pi%2Ax%2F%28x+-+1%29%29%29%29%29  is  -2  when  (x→ + ∞).

Solved.