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Question:

limx→2(√(1−cos2(x−2))/(x−2)) equals?

does not exist

equals −√2

equals √2

equals 1/√2

Solution:

Let x−2 = t ⇒ as x → 2, t → 0
So, limx→2(√(1−cos2(x−2))/(x−2)) = limt→0√(1−cos2t)/t = limt→0√(2sin²t)/t = limt→0√2|sint|/t
Now, limt→0−√2|sint|/t = −√2 and limt→0+√2|sint|/t = √2 ⇒ L.H.L ≠ R.H.L ⇒ limt→0√2|sint|/t does not exist.