# Tagged Questions

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### Bound for the integral over one region, given integrals over nearby regions

We have the following: $f:\mathbb{R}^2\rightarrow [0,1]$, and constants $b,c, \varepsilon_{x_k},\varepsilon_{y_k}$ such that: $$\tag{1}0<c<b$$ \tag{2} ...
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### Stacked with this Problem of Calculus

I have been struggling for quite some time with the following problem and I would really appreciate some help: Consider ...
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### Demonstrate that the limit of a function of two variables does not exist

From my multivariable textbook: $$\lim_{|x,y|\to|0,0|}\frac{y^2\sin^2 x}{x^4+y^4}$$ (original screenshot) Wolfram indicates that the limit DNE, but does not list the steps used to solve. Is there ...
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### multivariable calculus - find a function such that $\lim_{t \to 0} f(tx,ty)=0$

I was asked the following question: Find a function $f:\mathbb R^2 \to \mathbb R$ such that $f(x,y)$ has no limit as $x$ and $y$ approach zero, but $\lim_{t\to 0} f(tv)=0$ for all $v \in \mathbb R^2$ ...
### $L_{2}$ norm of the gradient of a vector valued function.
I have a vector valued function $U(x,y)=\Big(u_{1}(x,y),u_{2}(x,y)\Big)$. I want to find $\|\nabla U\|_{L_{2}(0,1)}$, but i could not figure how can do it. Do you have any idea?