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Limit questions

Evaluate the limit..

Lim x-->0  1 - cos x / x2


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1 Answer

If to read without parenthesis then the answer is (- ∞), because cos x can't be greater than 1 by its absolute value. I think your exression  should be written as

                                       lim x→0 (1-cos x)/x 2 (using parenthesis is inpoprtant to show the order of operations).

Now use trigonometric identities:           1- cos x = 2 sin2 (x/2)

You can prove it if you put x = 2 (x/2) and use the formula for cos of double angle, then

                                                          cos2 (x/2) = 1-sin2 (x/2) 

Thus, we can write

                                         Lim x→0 2sin 2 (x/2)/x 2 = 1/2  (answer)

because lim x→0 sin (x/2)/ (x/2) = 1