## anonymous 5 years ago Find an integer which is the limit of (1-cosx)/x as x goes to 0

1. anonymous

you can use the sandwitch thm

2. amistre64

1/x - cosx/x

3. anonymous

or you can do L'Hospitals rule

4. amistre64

hopitals rule was my thought :)

5. anonymous

if you take derivative of both top and bottom you'll get sin(x)

6. anonymous

so as x-->0, sin(x)-->0 so the answer is 0

7. amistre64

we can always do the long version :) 1-cos(x+h) - 1 + cos(x) --------------------- maybe? h

8. anonymous

thank you all. it's 0 as yuki said hehe

9. amistre64

its always been 0 lol

10. anonymous

is you use the sandwich thm $-1 \le \cos(x) \le 1$ $1 \ge -\cos(x) \ge -1$ $2 \ge 1-\cos(x) \ge 0$ ${2 \over x } \ge {1-\cos(x) \over x} \ge 0$

11. anonymous

now if you take the limit on both sides $\lim_{x \rightarrow 0} {2 \over x} \ge \lim_{x \rightarrow 0}{1-\cos(x) \over x} \ge \lim_{x \rightarrow 0}{0}$ $0 \ge \lim_{x \rightarrow 0}{1-\cos(x) \over x} \ge 0$ so the limit is 0