Evaluate the limit..
Lim _{x-->0 }1 - cos x / x^{2}_{ }
Evaluate the limit..
Lim _{x-->0 }1 - cos x / x^{2}_{ }
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 sin^{2} (x/2)
You can prove it if you put x = 2 (x/2) and use the formula for cos of double angle, then
cos^{2} (x/2) = 1-sin^{2} (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