Question

Find an integer which is the limit of (1-cosx)/x
as x goes to 0

Answer 1

you can use the sandwitch thm

Answer 2

1/x - cosx/x

Answer 3

or you can do L'Hospitals rule

Answer 4

hopitals rule was my thought :)

Answer 5

if you take derivative of both top and bottom
you'll get

sin(x)

Answer 6

so as x-->0, sin(x)-->0

so the answer is 0

Answer 7

we can always do the long version :)

1-cos(x+h) - 1 + cos(x)
--------------------- maybe?
h

Answer 8

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

Answer 9

its always been 0 lol

Answer 10

is you use the sandwich thm

\[-1 \le \cos(x) \le 1 \]

\[1 \ge -\cos(x) \ge -1\]

\[2 \ge 1-\cos(x) \ge 0\]

\[{2 \over x } \ge {1-\cos(x) \over x} \ge 0\]

Answer 11

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