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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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