Example 3.7. Is the function $z=$ $=f(x, y)=\sqrt{4-x^{2}-y^{2}}$ bounded above (below)?
Solution. The function $f$ is bounded from above and below, as $0 \leqslant f(x, y) \leqslant 2$. Moreover, the function has the maximum and minimum values on $D(f)$: $\max _{(x, y) \in D} f=2, \min _{(x, y) \in D} f=0$

## 3.2 .  LIMIT AND CONTINUITY

The definition of a limit is generalized for the case of functions of several variables as follows.

Let the function $f(P)$ be defined in some neighborhood of the point $P_{0}\left(x_{1}{ }^{0}, x_{2}{ }^{0}, \ldots, x_{n}{ }^{0}\right) \in D(f) \subset E^{n}$, except possibly at the point $P_{0}$. The function $f(P)$ has a limit at the point $P_{0}$, equal to $A$, if for any $\varepsilon>0$ there exists a $\delta(\varepsilon)>0$ such that for all points $P\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ in the domain of the function $D(f)$, for which $0<\rho\left(P, P_{0}\right)<\delta$, the inequality $|f(P)-A|<\varepsilon$ holds.

Notation

$$
\lim _{P \rightarrow P_{0}} f(P)=A ; \lim _{\rho\left(P, P_{0}\right) \rightarrow 0} f(P)=A ; \lim _{\substack{x_{1} \rightarrow x_{1}^{0} \\ x_{2} \rightarrow x_{2}^{0} \\ x_{n} \rightarrow x_{n}^{0}}} f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=A
$$

Theorems about limits of functions of one variable are easily generalized and remain valid for functions of several variables. It is important to note that the given definition of a limit assumes the simultaneous approach of the arguments $x_{1}, x_{2}, \ldots, x_{n}$ to their limits, that is, respectively, to $x_{1}^{0}, x_{2}^{0}, \ldots, x_{n}^{0}$. Therefore, the specified limit is also called an $n$-fold limit. Along with the $n$-fold limit, repeated limits are also considered, obtained as a result of limit transitions for each argument separately. In general, both types of limits of functions of several variables at a point can lead to different results, which must be taken into account when finding the $n$-fold limit of a function of variables.