[b]Parte (c)[/b][br]Si consideri la funzione [math]g\left(x\right)=2-\frac{3}{x}[/math] e si verifichi che [math]\forall x>0[/math] vale la disuguaglianza [math]f_n\left(x\right)>g\left(x\right)[/math], indipendentemente dal valore di [math]n[/math].[br]Si consideri l'integrale [math]I\left(t\right)=\int_1^t\left(f_n\left(x\right)-g\left(x\right)\right)dx[/math] che esprime l'area della regione delimitata dai grafici delle funzioni [math]f_n[/math] e[math]g[/math] e dalle rette di equazione [math]x=1[/math] e [math]x=t,t>1[/math].[br][br]Si calcolino [math]I\left(t\right)[/math] e il [math]lim_{_{t\longrightarrow+\infty}}I\left(t\right)[/math], fornendo un'interpretazione geometrica del risultato ottenuto.[br][br][b]Parte (d)[/b][br]Calcolare[br] [img]data:image/png;base64,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[/img][br]e verificare che il risultato non dipende da [math]n\in\mathbb{N},n>1[/math].[br]

[math]f_n\left(x\right)>g\left(x\right)[/math], cioè il grafico della famiglia di funzioni [math]f_n\left(x\right)[/math] si trova sempre al di sopra del grafico della funzione [math]g\left(x\right)=2-\frac{3}{x}[/math].[br]Infatti, semplificando la disequazione [math]2-\frac{3}{x}+\frac{3}{x^n}>2-\frac{3}{x}[/math] si ha [math]\frac{3}{x^n}>0[/math] che è vera [math]\forall n\in\mathbb{N},n>1[/math].

[math]I\left(t\right)=\int_1^t\left(f_n\left(x\right)-g\left(x\right)\right)dx=\int_1^t\frac{3}{x^n}dx=\left[-\frac{3}{\left(n-1\right)x^{n-1}}\right]_1^t=\frac{3}{n-1}\left(1-\frac{1}{t^{n-1}}\right)[/math][br][br][math]lim_{t\longrightarrow+\infty}\frac{3}{n-1}\left(1-\frac{1}{t^{n-1}}\right)=\frac{3}{n-1}[/math] [math]\forall n\in\mathbb{N},n>1[/math].[br][br]Utilizza gli slider nell'app di seguito per visualizzare l'area tra le due curve definita da [math]I\left(t\right)[/math] al variare di [math]n[/math] e [math]t>1[/math].[br]Sebbene l'area sia una porzione di piano illimitata a destra, ha una misura finita.

[img]data:image/png;base64,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[/img][br]Il limite non dipende da [math]n\in\mathbb{N}[/math] quando [math]n>1[/math]. [br]Numeratore e denominatore sono infiniti dello stesso ordine, ciascuno con infinito principale [math]x^n[/math].[br]Il limite è quindi il rapporto tra i coefficienti dei due infiniti principali a numeratore e denominatore.