26713: Difference between revisions

Latest revision as of 08:08, 8 January 2024

26713 (Radu Pop și Vasile Ienuțaș)

Se consideră șirul de numere reale Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_n)_{n \geq 0}} și $(y_{n})_{n\geq 0}$ cu $x_{n}\geq 1$ , $y_{n}\geq 1$ , pentru orice Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} , și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} (x_n^2 + y_n^2) = 2} . Să se calculeze Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} x_n} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} y_n} .
Soluție:

Avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \leq x_n - y_n \leq \sqrt{2(x_n^2 + y_n^2)} } și cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} \sqrt{2(x_n^2 + y_n^2)} = 2 } , rezultă că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} (x_n + y_n) = 2 } .

Cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \leq x_n \leq x_n + y_n - 1 } și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} (x_n + y_n - 1) = 1 } , obținem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} x_n = 1 } . Analog, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{{n \to \infty}} y_n = 1 } .

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-Avem <math> 2 \leq x_n - y_n \leq \sqrt{2(x_n^2 + y_n^2)} </math> și cum <math> \lim_{{n \to \infty}} \sqrt{2(x_n^2 + y_n^2)} = 2 </math>, rezultă că <math> \lim_{{n \to \infty}} (x_n + y_n) = 2 </math>. Cum <math> 1 \leq x_n \leq x_n + y_n - 1 </math> și <math> \lim_{{n \to \infty}} (x_n + y_n - 1) = 1 </math>, obținem <math> \lim_{{n \to \infty}} x_n = 1 </math>. Analog, <math> \lim_{{n \to \infty}} y_n = 1 </math>.
+Avem <math> 2 \leq x_n - y_n \leq \sqrt{2(x_n^2 + y_n^2)} </math> și cum <math> \lim_{{n \to \infty}} \sqrt{2(x_n^2 + y_n^2)} = 2 </math>, rezultă că <math> \lim_{{n \to \infty}} (x_n + y_n) = 2 </math>.
+Cum <math> 1 \leq x_n \leq x_n + y_n - 1 </math> și <math> \lim_{{n \to \infty}} (x_n + y_n - 1) = 1 </math>, obținem <math> \lim_{{n \to \infty}} x_n = 1 </math>. Analog, <math> \lim_{{n \to \infty}} y_n = 1 </math>.