Articles of calculus

taking the limit of $f(x)$, questions

How do I take the limit of the following functions? I had included some of my thoughts with them. $\lim_{x\to\infty}\dfrac{4x^3 – 2x + 1}{8x^3 + \sin(x^2) – x^{-1}}$; my thoughts: I am not sure about the bottom since there are the sine function and $-1$ power $\lim_{x\to\infty}\dfrac{e^x}{x^{x-1}}$; my thoughts: isn’t $e^x$ faster since $x^{x-1}$ is […]

Green's function of a bounded domain is strictly negative

I have been struggling with this proof for half a day. I am really baffled. The result of my proof is close but not exactly the same as the proposal. Here is the question: Here is my proof: The proposal is $G<0$, but what my proof arrives at $G\le0$. Where did I go wrong? Thanks […]

How to find $\lim_{(x,y)\rightarrow (0,0)} \frac{\sin(x\cdot y)}{x}$?

$$\lim_{(x,y)\rightarrow (0,0)} \frac{\sin(x\cdot y)}{x}$$ How can I find this limit?

Exterior derivative of a 2-form

I want to prove that the exterior derivative of a 2-form in $\mathbb{R}^n$: $${\alpha = \sum a_{ij} dx_i \wedge dx_j}$$ is: $${d \alpha = \sum (\frac{\partial a_{ij}}{\partial x_k} + \frac{\partial a_{jk}}{\partial x_i} – \frac{\partial a_{ik}}{\partial x_j})dx_i \wedge dx_j \wedge dx_k.}$$ The thing is that I’m stucked after taking the differential of the function $a_{ij}$ and […]

Proving the existence of a sequence of polynomials convergent to a continuous function $f$.

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. I would appreciate any help/direction …

Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $

How does $$ \prod_{k\ =\ 1}^{\infty}\left(\,\,\sqrt{\, k+1 \over k\,}\,{k \over k + 1/2}\,\right) ={\,\sqrt{\,\pi\,}\, \over 2} ={\sqrt{2\left(\,\pi/2\,\right)} \over 2} ={1 \over 2}\,\,\sqrt{\, 2\prod_{k\ =\ 1}^{\infty} {\left(\, 2k\,\right)^{2} \over \left(\, 2k – 1\,\right)\left(\, 2k + 1\,\right)}\,}\,\ {\large ?} $$ In other words, how do you derive the Wallis product ${\pi \over 2} =\prod_{k\ =\ 1}^{\infty} {\left(\, […]

Finding the equation of the normal line

I have a question to find the equations of the tangent line and the normal line to the curve at the given point. I can find the equation for the tangent line easily but I am not sure what a normal line is and there is no example that I can find. $y=x^4 + 2e^x$ […]

How to do contour integral on a REAL function?

Suppose we are given the problem: Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} dx$$ Where $x$ is a real variable. A real variable function (no complex variables). I was reading Schaum’s outline to this solution and it defines: $f(z) = \displaystyle \frac{1}{z^6 +1}$ And it says consider a closed contour $C$ Consisting of the line from $-R$ […]

Prove that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$

Hi everyone I have been trying to prove that that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$ . Heres my attempt: LS: $ \int \limits_a^b f(x) dx$ = $ \int \limits f(b) – \int \limits f(a) $ RS: $ \int \limits_a^b f(a+b-x) dx$ = $ \int \limits f(a+b-b) – \int \limits […]

Interesting problem of finding surface area of part of a sphere.

Show that the surface area of a zone of a sphere that lies between two parallel planes is $2\pi Rh$, Where $R$ is the radius of the sphere and $h$ is the distance between the planes. If you are wondering what is interesting about this ? The fact that the surface area depends only on […]