Question

determine the following limit:
lim θ -> 0, lim (θ->0) (θ^2-3 sin(θ))/θ

Answer 1

If I understand you rightly, this is the limit you must determine:\[\lim_{\theta \rightarrow 0}\frac{ \theta^2-3\sin \theta }{ \theta }\]I'd split it up:\[=\lim_{\theta \rightarrow 0}\frac{ \theta^2 }{ \theta }-3\lim_{\theta \rightarrow 0}\frac{ \sin \theta }{ \theta }\]The first limit is, after simplifying, just θ going to 0. Result 0.
The second one is a standard limit, which you might know...

Answer 2

\[\lim_{\theta \rightarrow 0}\frac{ \sin \theta }{ \theta }=1\]
This is a standard limit that normally may be used ;)
So the limit you have to calculate is:

0 - 3*1 = -3