Question

Trig limits and Squeeze Theorem

lim (theta cos(1/theta) as theta approaches 0

http://www.wolframalpha.com/input/?i=lim+%28theta+cos%281%2Ftheta%29%29+theta+approaches+0

Why does this equal 0?
The steps I followed were to take: lim(theta) as theta approaches 0. This equals 0.
Then lim (cos 1/theta) is undefined? If this isn't a finite number, how does it end up being 0? 0 * a non finite number doesn't equal 0?

Answer 1

it is zero because \(\cos(\theta)\) is stuck between \(-1\) and \(1\) and therefore
\[\theta \cos(\theta)\] is stuck between \(-\theta\) and \(\theta\)
as \(\theta \to 0\) then you have to get \(0\)