📝 题目
5. 1.25 设 $f\left( {x,y}\right)$ 除直线 $x = a$ 与 $y = b$ 外有定义,且满足:
(1) $\mathop{\lim }\limits_{{y \rightarrow b}}f\left( {x,y}\right) = \varphi \left( x\right)$ 存在;
(2) $\mathop{\lim }\limits_{{x \rightarrow a}}f\left( {x,y}\right) = \psi \left( y\right)$ 一致存在 (即 $\forall \varepsilon > 0,\exists \delta \left( \varepsilon \right) > 0$ ,当 $0 < \left| {x - a}\right| < \delta$ 时, $\forall y \neq b$ ,有 $\left| {f\left( {x,y}\right) - \psi \left( y\right) }\right| < \varepsilon$ ).
证明:
(1) 累次极限 $\mathop{\lim }\limits_{{x \rightarrow a}}\mathop{\lim }\limits_{{y \rightarrow b}}f\left( {x,y}\right) = \mathop{\lim }\limits_{{x \rightarrow a}}\varphi \left( x\right) = c$ 存在;
(2)累次极限 $\mathop{\lim }\limits_{{y \rightarrow b}}\mathop{\lim }\limits_{{x \rightarrow a}}f\left( {x,y}\right) = \mathop{\lim }\limits_{{y \rightarrow b}}\psi \left( y\right) = c$ ；
(3) 全面极限 $\mathop{\lim }\limits_{{\left( {x,y}\right) \rightarrow \left( {a,b}\right) }}f\left( {x,y}\right) = c$ .
💡 答案与解析
5. 1.25 (1) 先证 $\forall \varepsilon > 0,\exists \delta > 0$ ,当 $0 < \left| {{x}_{1} - a}\right| < \delta ,0 < \left| {{x}_{2} - a}\right| < \delta$ 时, 对 $\forall y \neq b$ ,有 $\left| {f\left( {{x}_{1},y}\right) - f\left( {{x}_{2},y}\right) }\right| < \varepsilon$ ;
(2)利用
$$ \left| {\psi \left( y\right) - c}\right| \leq \left| {\psi \left( y\right) - f\left( {{x}_{1},y}\right) }\right| + \left| {f\left( {{x}_{1},y}\right) - \varphi \left( {x}_{1}\right) }\right| + \left| {\varphi \left( {x}_{1}\right) - c}\right| . $$
先取 ${x}_{1}$ 充分接近于 $a$ ,使前后两个绝对值小于 $\frac{\varepsilon }{3}$ ,然后固定 ${x}_{1}$ ,找 $\delta$ ,使得当 $0 <$ $\left| {y - b}\right| < \varepsilon$ 时,中间那个绝对值也小于 $\frac{\varepsilon }{3}$ .