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Problem on limits

B.Sudhakar
22/01/2017 0 0

Let's solve a problem on limits

Find limit:    x→0  (1 - cos2x ) /(x²)

On direct substitution we get 1 - 1/0 = 0/0 which is in indeterminate form

Let's substitute (1 - cos 2x ) = 2 sin² x 

= limit x→0  (2 sin² x )/( x² )

= limit x→0 [ 2 ( sin x / x )² ]

= 2 [ limit x→0 (sin x / x) • limit x→0 ( sin x / x ).        { limit x→0 (sin x / x) = 1}

= 2 [ 1 • 1 ] = 2

∴ limit x→0 ( 1 - cos 2x ) / x² = 2.

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