The limit of (xcsc2x)/(cos5x) as x approaches 0 can be evaluated using the fundamental limit lim as x approaches 0 (sinx)/x = 1. By rewriting the expression as (x(1/sin2x))/(cos5x), one can apply the limit properties. Specifically, the limit lim as x approaches 0 of (2x)/(sin(2x)) equals 1, leading to the conclusion that the overall limit evaluates to 0, since cos(5x) approaches 1 as x approaches 0.
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