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torquerotates
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Homework Statement
Prove tan(x) is unbounded on [0,pi/2)
Homework Equations
if s is unbounded, s>nThe Attempt at a Solution
So I'm doing this by constructing a sequence. I don't know if this is valid. If tan(x) is unbounded on [0,pi/2), there exists x(n) in [0, pi/2) such that |tan(x(n))|>n for all n.
I can construct an sequence on [0,pi/2) such that pi/4<x1<x2<x3<...<x(n). such that
for n=1, tan(x1)>1
sin(x1)/cos(x1)>1 true since pi/4<x1<=pi/2
assume tan(x(n))>n is true.
well tan(x(n+1))=tan(e+x(n)) { since x(n+1)>x(n), I can say, x(n+1)=x(n)+e
for some number e}
=[tan(e)+ tan(x(n))]/[1-tan(e)tan(x(n)]<[tan(e)+n]/[1-tan(e)tan(x(n)]
taking lim[tan(e)+n]/[1-tan(e)tan(x(n)]=(0+n)/[1-0]=n
e->0
hence tan(x(n+1))>n
hence tan(x) is unbounded on [0,pi/2)
I don't even know if this is correct. I don't know if I can construct a sequence that does this and I don't know if I can take limits that way.