You want the limit of y=(1-cos 2x)/x^2 while x approaches 0.

y = (1-cos 2x)/x^2

=> [1 - (1 - 2*(sin x)^2)]/x^2

=> 2*(sin x)^2/x^2

=> 2*(sin x / x)^2

lim x--> 0 (sin x / x) = 1

Using this identity.

lim x--> 0 [ (1-cos 2x)/x^2]

=> lim x--> 0 (2*(sin x/x)]

=> (2)* lim x--> 0 [(sin x/x)]

=> 2*1

=> 2

**The required limit is 2.**

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