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Explications étape par étape:
To find the limit of the expression as x approaches π/2, we can first substitute π/2 for x:
lim(x→π/2) [(2x_π)/sin(cos(x))]
= [(2(π/2)_π)/sin(cos(π/2))]
= [(π)_π/sin(cos(π/2))]
= [(π)_π/sin(0)]
Now, since sin(0) equals 0, and division by 0 is undefined, we need to approach π/2 from both the left and the right to determine if the limit exists.
Approaching from the left:
sin(0) = 0, so [(π)_π/sin(0)] approaches [(π)_π/0], which is undefined.
Approaching from the right:
sin(0) = 0, so [(π)_π/sin(0)] approaches [(π)_π/0], which is also undefined.
Since the limit is undefined from both sides, the overall limit does not exist.