I know to how prove normal limits using the epsilon-delta definition, say: $$\lim_{x\to a}f(x) = L$$ But, there was a question on my textbook which I couldn’t quite figure out to do, even though I’ve thought about it for a while I don’t even know how to go about starting it. Use $\varepsilon$-$\delta$ definition of […]

Given the following limit: $$ \lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3} $$ And the instrucion to “Determine whether the limit exists, give a complete argument”, would the following be a “complete argument”? Approaching the limit from the line y=0, gives $ \lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3} = \lim_{(x,y)\to (0,0)} \frac{0}{x^3+y^3} = 0 $ Approaching the limit from the line y=x, […]

Consider the limit product rule: $$\lim_{x\rightarrow c} (f(x)⋅g(x))=[\lim _{x\rightarrow c} f(x)]⋅[\lim_{x\rightarrow c} g(x)]$$ Now consider, for the sake of the argument, $f(x) = x, g(x) = (e/x)$ Clearly, the limit is e. However, by the product it would be impossible to figure out. Does this mean that the product rule is only valid when the […]

What do we mean by Indeterminate form ? can we show $0/0$ anything as we wish i mean is it not unique there can be several answers for it

If the limit of one sequence $\{a_n\}$ is zero and the limit of another sequence $\{b_n\}$ is also zero does that mean that $\displaystyle\lim_{n\to\infty}(a_n/b_n) = 1$?

I’m trying to determine all subsequential limit points of the following sequence X_n = cos(n) Not sure how to decompose this into subsequences. Anyone know how to tackle this problem? Thanks!

This question is an exact duplicate of: Precise definition of limits and $\lim_{x\to-1}\frac1{\sqrt{x^2+3}}=\frac12$ 1 answer

Is there an easy way to calculate $$\lim_{k \to \infty} \frac{(k+1)^5(2^k+3^k)}{k^5(2^{k+1} + 3^{k+1})}$$ Without using L’Hôpital’s rule 5000 times? Thanks!

Here is my question: Does the following limit exist? $$ \lim_{y\to\xi}\frac{(\xi_i-y_i)(\xi_j-y_j)({{\xi}-y})\cdot n(y)}{|\xi-y|^5},\quad 1\leq i,j\leq 3,\tag{*} $$ where $S\subset{\mathbb R}^3$ is a surface which has a continuously varying normal vector, $\xi=(\xi_1,\xi_2,\xi_3)\in S$, $y=(y_1,y_2,y_3)\in S$, $n(y)$ is the [EDITED: unit] normal vector at point $y$. Here $(\xi-y)\cdot n(y)$ is the dot product. In the spirit of Polya, […]

1) Let $q\in(0,1)$ is fixed and $L$ is a finite value. Is it possible to say if $\lim_{x\to\infty}f(qx)=L$ then $\lim_{x\to\infty}f(x)=L.$ 2) And i also stack in if for all $\epsilon>0$ is it possible to say if $L-\epsilon\leq f(\frac{x}{q})$ and $f(qx)\leq L+\epsilon$ then $\lim_{x\to\infty}f(x)=L.$

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