Question

lim cosx-1 / x

x->0

Answer 1

Imitate the trick we just used in the your last problem. Multiply top and bottom by
(cos x + 1).

Remember also that
\[\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1\]
a result your teacher should have given to you or proved in class.

Answer 2

actually you have to apply expansion formula of cosx
which is 1- x^2/(2!) + x^4/4! - .. .......
then cosx -1 will be x^2/(2!) + x^4/4! - .. .......
now lim (cosx -1)/x = lim {x/2! ..........}
it is zero......because each term contains x

Answer 3

\[\lim_{x \to 0}\frac{\cos(x)-1}{x}=\lim_{x \to 0}\frac{\sin(x)}{x}\frac{\cos(x)-1}{\sin(x)}= \cdots\]

Answer 4

then use the result from your last problem

Answer 5

you have another way also
cosx-1=-2sin^2 (x/2)
if you use this you still find zero

Answer 6

so the answer is zero?

Answer 7

yes

Answer 8

i'm so confused with everything that was written. but i got zero for my answer the way i did it

Answer 9

@Zarkon: nice

@Prashant: definitely not a good idea in this context, as you're making the problem more complex. Also, technically, you would need to show that that series does converge to zero; it's not enough to observe that you can put x = 0 everywhere; so definitely more complicated.