## Limit of $x\ln{x}$

I am trying to solve $$\lim \limits_{x \to 0}x\ln{x}$$ which according to WolframAlpha (and Wikipedia) equals $0$. I managed to solve it by substituting such that $y = \dfrac{1}{x}$ and then using

mathematics calculus limits
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I am trying to solve $$\lim \limits_{x \to 0}x\ln{x}$$ which according to WolframAlpha (and Wikipedia) equals $0$. I managed to solve it by substituting such that $y = \dfrac{1}{x}$ and then using

mathematics calculus limitsIn Allen Hatcher's Algebraic Topology, $X\vee Y$ means the "wedge sum" of two (topological) spaces $X$ and $Y$. However, in $\LaTeX$, \wedge is the notation for $\wedge$, while $\vee$ is

mathematics notationI understand the concept of Center Of Mass(com), but I am having a difficult time interpreting the equation of the simplified case of one-dimension. The book I am reading defines the position of t...

physics newtonian-mechanics mass particles dimensional-analysis densityHow can I evaluate this limit: $$\lim_{x\to\infty}x\left(\frac\pi2-\arctan x\right).$$ I know that the correct answer is $1$, but why?

mathematics calculus limitsThe question is from Hatcher's Algebraic Topology Problem 2.2.24. *Suppose we build $S^2$ from a finite collection of polygons by identifying edges in pairs. Show that in the resulting CW structu

mathematics algebraic-topologyI apologize if this question has been asked. I know several similar ones have been asked but I cannot find one answering this in particular. I want to know what this summation means:

mathematics notationIn a graph, there are points that need to be visited. For each of these points, there is a certain time interval given by its start and end times, telling the time interval during which that point...

mathematics graph-theory algorithms np-completeI'm trying to evaluate Euler Lagrange equation from the following relation: $$ F[f(\vec{r})]=\int_{\vec{r_1}}^{\vec{r_2}} d^n r J[f(\vec{r}),\nabla f (\vec{r}),H f(\vec{r}) ] $$ where $H$ is the

mathematics calculus-of-variations euler-lagrange-equationFor example, should I write : (1) Chris's dog OR (2) Chris' dog (1) the infants's toy OR (2) the infants' toy ? I read in Struck & White (4th edition from 1979) that you shoul

english-language possessiveThe following quotation is a line from Ron to Harry after the first stage of the Triwizard Tournament. (p359, Harry Potter 4, US edition) “You were the best, you know, no competition. Cedric di.

english-language meaning articles determinersI have been working on this optimization problem for days but I cannot figure out the right way to finish it off. I am reading from Optimization Theory by Pierre, and this is problem 3.3. Note that

mathematics optimization calculus-of-variationsI'm looking at a derivation of the Euler-Lagrange equations in a field setting, and one step in the proof is continually eluding me. Let $\phi(\vec x,t)$ be a field and $\mathscr

mathematics mathematical-physics calculus-of-variations euler-lagrange-equationUnder the Linear Blurring Model - $ f = H \ast u $. I'm trying to calculate the Euler Lagrange of with respect to $ u $ of the functional: $$ E \left( u \right) = {\left\| f - H \ast u \right\|}^

mathematics calculus optimization calculus-of-variationsFor learning purposes, I'm trying to prove that the shortest function passing through the two points $(x_1, y_1)$, $(x_2, y_2)$ is a straight line, without using the Euler-Lagrange equation. My a

mathematics optimization functional-equations calculus-of-variations euler-lagrange-equationI'm working on a problem in computer vision and I've ended up trying to minimize the functional $$\int \left[\lambda(S''(x))^2 + (f(x) - S(x))^2 \sum_k \delta (x - x_k)\right]dx$$ wh

mathematics differential-equations multivariable-calculus calculus-of-variations euler-lagrange-equationThis is a homework question so giving the full answer is not the intention. Rather, I am looking for a hint. I am asked to minimise the functional $F[u] = \int_\Omega\: \frac{1}{2} E(x,y) \left( \...

mathematics euler-lagrange-equationI've been able to derive the Euler-Lagrange equation for $$\int_a^b F(x,y,y')dx$$ relatively easily by using the total derivative and integration by parts. However, I was unable to apply the

mathematics differential-equations multivariable-calculus calculus-of-variations euler-lagrange-equationI wonder what is the alternative to Euler-Lagrange equations when we have non-differentiability issues. I'll give an example: Dido's problem can be stated as: Find the figure bounded by

mathematics calculus-of-variations lagrange-multiplier euler-lagrange-equationHow does one see that that the Euclidean AdS is the same as the hyperbolic space at the same dimension ie $EAdS_n = \mathbb{H}_n = SO_0(n,1)/SO(n)$? Or is this to be seen as the definition of

physics quantum-field-theory string-theory ads-cft topology anti-de-sitter-spacetimeSuppose $u$ is a solution of the two-dimensional wave equation and that at $t=0, u=u_t=0$ outside the disc $x_1^2+x_2^2 \le 1$. Up to what time can you be sure that $u=0$ at the point $(x_1,x_2)=(2...

mathematics analysis pde wave-equationI'm reading a paper in which it gives the following Lagrangian $$L[u,\rho,\phi]=L_0[u,\rho]+\phi(x)(\partial_t\rho+\nabla\cdot(\rho u))$$ where $L_0$ is part of Lagrangian and $\phi(x)$ is Lagrang

mathematics multivariable-calculus partial-derivative calculus-of-variations lagrange-multiplierSpose $ u $ solves the wave equation on $ U \subset \mathbb{R}^3 $ with initial conditions $ u (x, 0) = g(x)$ and $ u_t(x,0) = h(x)$, where lower script indicates partial differentiation. Then we h...

mathematics integration analysis pde wave-equationA necessary and sufficient condition for the calculus of variations problem $\delta\int_{a}^{b} L(x(t), \frac{dx}{dt}) dt = 0$ be independent of the choice of parametric representation of the curve

mathematics calculus-of-variationsIt is said in Weinberg's Book, Gravitation and Cosmology, page 377, that a killing vector field (which we a priori assume exists globally) can be uniquely determined by its value and first derivat

physics general-relativity differential-geometry vector-fields