Limit questions

Question

Evaluate the limit..

Lim _x-->0  1 - cos x / x²

 

Answer 1

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