Question

(6)
Evaluate the lim
x tends to x/2
(1- sinx / sin x + cos 2x ) ​

Answer 1

Answer:

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Answer 2

Answer:

Let us multiply the numerator and denominator of (1 - cos x) / x by (1 + cos x) and write

limx→0 (1 - cos x) / x

= limx→0 [ (1 - cos x) / x ] *[ (1 + cos x) / (1 + cos x) ]

The numerator becomes 1 - cos 2 x = sin 2 x, hence

limx→0 (1 - cos x) / x

= limx→0 [ (sin 2 x) / x ] *[ 1/ (1 + cos x) ]

The limit can be written

limx→0 (1 - cos x) / x

= limx→0 [ (sin x) / x ] * limx→0 [ sin x / (1 + cos x) ] = (1)(0/2) = 0

We have used the theorem: limx→0 [ (sin x) / x ] = 1