anonymous
  • anonymous
Find an integer which is the limit of (1-cosx)/x as x goes to 0
Mathematics
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anonymous
  • anonymous
Find an integer which is the limit of (1-cosx)/x as x goes to 0
Mathematics
chestercat
  • chestercat
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yuki
  • yuki
you can use the sandwitch thm
amistre64
  • amistre64
1/x - cosx/x
yuki
  • yuki
or you can do L'Hospitals rule

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amistre64
  • amistre64
hopitals rule was my thought :)
yuki
  • yuki
if you take derivative of both top and bottom you'll get sin(x)
yuki
  • yuki
so as x-->0, sin(x)-->0 so the answer is 0
amistre64
  • amistre64
we can always do the long version :) 1-cos(x+h) - 1 + cos(x) --------------------- maybe? h
anonymous
  • anonymous
thank you all. it's 0 as yuki said hehe
amistre64
  • amistre64
its always been 0 lol
yuki
  • yuki
is you use the sandwich thm \[-1 \le \cos(x) \le 1 \] \[1 \ge -\cos(x) \ge -1\] \[2 \ge 1-\cos(x) \ge 0\] \[{2 \over x } \ge {1-\cos(x) \over x} \ge 0\]
yuki
  • yuki
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

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