Iterated and repeated limits of several-variable functions

Prove that for the function

we have

so that $\lim \limits_{\substack {x \to 0 \\ y \to 0}} f(x,y)$ does not exist.

Prove that for the function

we have

yet $\lim \limits_{\substack {x \to 0 \\ y \to 0}} f(x,y)$ does not exist.

Prove that for the function

the iterated limits $\lim \limits_{x \to 0}\left\{ \lim \limits_{y \to 0} f(x,y)\right\}$ and $\lim \limits_{y \to 0}\left\{ \lim \limits_{x \to 0} f(x,y)\right\}$ do not exist, yet $\lim \limits_{\substack {x \to 0 \\ y \to 0}} f(x,y)=0$.

Find $\lim \limits_{x \to a}\left\{ \lim \limits_{y \to b} f(x,y)\right\}$ and $\lim \limits_{y \to b}\left\{ \lim \limits_{x \to a} f(x,y)\right\}$, where:

Show that

Show that

while the quantity

nevertheless has no meaning.

Show that $f(x)$, defined by the equality

equals one if $x$ is a rational number, and zero if $x$ is an irrational number; $m$ and $n$ are natural numbers.

Find:

1. $\lim_{x \rightarrow 0} \lim_{y \rightarrow 0} u$;

2. $\lim_{y \rightarrow 0} \lim_{x \rightarrow 0} u$;

3. $\lim_{\substack {x \rightarrow 0 \\ y \rightarrow 0}} u$; if:

Construct a function $u = f(x ; y)$, defined on the whole plane, for which:

Let the function $f$ be defined on a set $E$ containing a neighborhood of the point $\left(x_{0} ; y_{0}\right):\left|x-x_{0}\right| < \delta_{1},\left|y-y_{0}\right| < \delta_{2}$, except possibly for the points of the lines $x = x_{0}$ and $y = y_{0}$. Prove that if $\lim_{\substack {x \rightarrow x_{0} \\ y \rightarrow y_{0}}} f = A$ and for every $y \in \left(y_{0}-\delta_{2} ; y_{0}+\delta_{2}\right), y \neq y_{0}$, the limit $\lim_{x \rightarrow x_{0}} f$ exists, then $\lim_{y \rightarrow y_{0}} \lim_{x \rightarrow x_{0}} f = A$.

Find:

1. $\lim_{x \rightarrow \infty } \lim_{y \rightarrow \infty } u$;

2. $\lim_{y \rightarrow \infty } \lim_{x \rightarrow \infty } u$;

3. $\lim_{\substack {x \rightarrow \infty \\ y \rightarrow \infty }} u$; if:

Given the function $u = \frac{x^{y}}{1+x^{y}}$.

Let $u_{m, n}(x) = \cos^{m}(\pi n ! x) = \cos^{m}(\pi n ! x), m, n \in N, x \in R$. Determine for which values of $x$ the equality

holds.

Let $b_{m, n}, m, n=1,2, \ldots$, be a double sequence of real numbers, which for each fixed $m$ is nondecreasing in $n$. Show that $\lim_{n \rightarrow \infty } \lim_{m \rightarrow \infty } b_{m, n}=\lim_{m, n \rightarrow \infty } \inf b_{m, n}$ and $\lim_{m \rightarrow \infty } \lim_{n \rightarrow \infty } b_{m, n}=\lim_{m, n \rightarrow \infty } \sup b_{m, n}$ provided the indicated limits exist (they may be infinite) and where $\lim \inf b_{m, n}$ and $\lim \sup b_{m, n}$ denote, respectively, the smallest and the largest limit points attainable by a sequence $b_{m_{k}, n_{k}}, k=1,2, \ldots$, with $m_{k} \rightarrow \infty$ and $n_{k} \rightarrow \infty$.