Question

prove that limit approaches x to 0 sine delta x/delta x=1

Answer 1

I assume you mean that you want to prove the following: \[\lim_{\Delta x \rightarrow 0} \frac{ \sin \Delta x }{ \Delta x } = 1\] This can be done using L'Hospital's Rule or the Squeeze Theorem.
http://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx shows a proof using the Squeeze Theorem.