Articles of limits

Use the $\varepsilon$-$\delta$ definition of a limit to prove this.

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 […]

Determining existence of limit with multiple variables: $\lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3}$

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, […]

What are the limitations of the limit product rule?

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 […]

Indeterminate form from calculus

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

Sequences that tend to zero

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 $\displaystyle\lim_{n\to\infty}(a_n/b_n) = 1$?

Subsequential limit of sequence

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!

Easier method to prove limit by epsilon delta definition.

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 $\displaystyle\lim_{k \to \infty} \frac{(k+1)^5(2^k+3^k)}{k^5(2^{k+1} + 3^{k+1})}$?

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!

How should I deal with this two-dimensional $\frac{0}{0}$ limit?

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, […]

Limit properties as $x\to \infty$ for functions

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 $\lim_{x\to\infty}f(x)=L.$