Question

Help required.... :(
lim (1-sin theta)/ theta and theta approaches to 0

Answer 1

\[\lim_{\theta \rightarrow 0}\frac{ 1-\sin \theta }{ \theta } \]

Answer 2

Hint: \(\lim\limits_{x\rightarrow0}\dfrac{\sin x}x=1\)

Answer 3

in this case we need to consider both left hand limit and right hand limit
know what those are ?

Answer 4

According to the limit theorem for differences, Amtal's original limit can be broken into two separate limits: [lim as theta approaches zero of 1/theta] minus [lim as theta approaches zero of (sin theta)/theta]. The second limit should now be obvious. Hart is correct in that we need to consider both the left and right hand limits as theta approaches zero. Try it.

Answer 5

So according to mathmale and hartnn guidance and suggestion, the two-side limits should be as following, you may try to draw y = 1/x to find out why
\[\lim_{\theta \rightarrow 0^{+}} (\frac{ 1 }{ \theta } - 1) = +\infty\] and \[\lim_{\theta \rightarrow 0^{-}} (\frac{ 1 }{ \theta } - 1) = -\infty\]