How can I study the limit of the following function? $$\begin{equation*} \lim_{x \rightarrow 0} \frac{[x]}{\sin x} \end{equation*}$$ Any hint will be appreciated! Thanks!

I am interested in integrals of the form $$ I_n(r) = \int_0^rdr_1\cdots\int_0^rdr_n\,\Theta_A|\mathbf{H}_1\times\mathbf{H}_2|(\mathbf{H}_3\cdot\mathbf{H}_4)\cdots(\mathbf{H}_{2n-1}\cdot\mathbf{H}_{2n}), $$ where $\mathbf{H} = (a,b)$ is a two-dimensional vector, $\cdot$ denotes the Euclidean inner...

Find with complete proofs the interior points of the following set: $$S = {(x,y)\in \mathbb{R}| |x+y|<1}$$ I see that all the points of the set are interior, but how can I prove this? Any help will be appreciated! thanks!

Find with complete proofs the exterior points of the following set: $$S = \{(x,y)\in \mathbb{R}| |x+y|<1\}$$ I see that the exterior points of the set are $|x+y|>1$ , but how can I prove this? could anyone help me please? thanks!

Suppose the ray AD is in the interior of the $\angle BAC$, and the ray AE is in the interior of the $ \angle DAG$. Show that AE is also in the interior of $ \angle BAC$. Using Hilberts axioms this is exercise 9.2 in Hartshorne pg 96. I have alot of trouble with proving rigorously really obvious ...

Show that "halves of equals are equal" in the following sense: if $AB \cong CD$, and if E is a midpoint of AB in the sense that $A * E * B $ and $AE \cong EB$, and if F is a midpoint of CD, then $AE \cong CF.$ Conclude that a midpoint of AB, if it exists, is unique. I am not really sure what t...