Esercizi di matematica per le superiori/Limiti

Template:Esercizi di matematica per le superiori Richiami di teoria: Limiti. <quiz display=simple> { Seleziona i limiti al finito corretti e, dopo averli corretti, trascrivili su un foglio e dimostra che questi limiti sono corretti: |type="[]"} + $\underset{x\to -3}{\lim }\frac{x+3}{1+x^2}=0$ + $\underset{x\to 8}{\lim }{\log }_2(x+8)=4$ - $\underset{x\to -1}{\lim }\frac{2x^2+5x+3}{4x^2-3x+2}=3$ + $\underset{x\to 1}{\lim }(x^2+x+8)=10$ - $\underset{x\to \frac{\pi }{3}}{\lim }\frac{\sin 2x+\cos x}{\sin 5x-\cos (2x-\frac{\pi }{3})}=1$ - $\underset{x\to 3}{\lim }(x^3+x+1)=30$ + $\underset{x\to 0}{\lim }\frac{e^x}{2^x}=1$ - $\underset{x\to \pi }{\lim }\frac{\tan x+\cos x}{\cos x+\sin 2x}=0$ - $\underset{x\to 2}{\lim }\frac{x^2+4}{2x-4}=2$ + $\underset{x\to \frac{1}{2}}{\lim }\frac{2x^3+x^2-2x-1}{4x^2-1}+\frac{5x}{4}+x=1$

{ Calcola i seguenti limiti al finito di queste funzioni: |type="{}"} $\underset{x\to 1}{\lim }-\frac{x+3}{x-2}$ { 1 } $\underset{x\to 0}{\lim }{\log }_2(x+8)$ { 3 } $\underset{x\to -1}{\lim }\frac{x^2-3x+2}{x^2-7x+12}-\frac{7x}{10}$ { 1 } $\underset{x\to 0}{\lim }{\log }_2(8^x+x^8+8)$ { 3 } $\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin x+\cos x}{\sin 2x-\cos (x-\frac{\pi }{2})}$ { 1 }

{ Calcola i seguenti limiti all'infinito di queste funzioni: |type="{}"} $\underset{x\to +\mathrm{\infty }}{\lim }(-x+3)$ { - infinito } $\underset{x\to +\mathrm{\infty }}{\lim }{\log }_2(x+8)$ { + infinito } $\underset{x\to -\mathrm{\infty }}{\lim }\frac{x^2-3x+2}{x^2-7x+12}-\frac{7x}{10}$ { + infinito } $\underset{x\to \mathrm{\infty }}{\lim }\frac{3x^5-3x+2}{12-x^2+3x^5+x+x^3}$ { 3 }

{ Calcola i seguenti limiti al finito di queste funzioni: |type="{}"} $\underset{x\to 1}{\lim }\frac{x^3-x^2-x+1}{x^3+x^2-x-1}$ { zero } $\underset{x\to \mathrm{\infty }}{\lim }\frac{x^3+x}{x^4-3x^2+1}$ { zero } $\underset{x\to 1}{\lim }(\frac{1}{1-x}-\frac{3}{1-x^3})$ { - 1 } $\underset{x\to 1}{\lim }\frac{x^2-\sqrt{x}}{\sqrt{x}-1}$ { 3 } $\underset{x\to +\mathrm{\infty }}{\lim }\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}$ { 1 } $\underset{x\to 0}{\lim }\frac{\sqrt{1+x+x^2}-\sqrt{1-x+x^2}}{x^2-x}$ { - 1 } $\underset{x\to 1}{\lim }\frac{\sqrt{x+3}-\sqrt{3x+1}}{x-1}$ { - 1/2 } $\underset{x\to 0}{\lim }\frac{\sqrt{1+x^2}-1}{x^2}$ { 1/2 } $\underset{x\to 0}{\lim }\frac{\sin x}{|x|}$ { +- 1 } $\underset{x\to 0}{\lim }\frac{\sqrt{1+x\sin x}-1}{x^2}$ { 1/2 } $\underset{x\to \frac{\pi }{4}}{\lim }[(1-\tan x)\tan 2x]$ { 1 } $\underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{1-\cos x^3}$ { 2/3 } $\underset{x\to 0}{\lim }\frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}$ { zero } $\underset{x\to 0}{\lim }\frac{\sin 2x}{\ln (1+x)}$ { 2 } $\underset{x\to 0}{\lim }\frac{\sin 3x-\sin x}{\ln (1+x)}$ { 2 } $\underset{x\to 0}{\lim }\frac{\ln (x+2)-\ln 2}{x}$ { 1/2 } $\underset{x\to 0}{\lim }\frac{\ln (\cos x)}{x^2}$ { - 1/2 } $\underset{x\to \mathrm{\infty }}{\lim }{\left(\frac{x^2+2}{x^2}\right)}^{x^2+2}$ { e^2 } $\underset{x\to 1}{\lim }{\left(\frac{x-1}{x^2-1}\right)}^{x+1}$ { 1/4 } $\underset{x\to \mathrm{\infty }}{\lim }{\left(\frac{x^2+2}{2x^2+1}\right)}^{x^2}$ { zero } $\underset{x\to 0}{\lim }{(\cos x)}^{\frac{1}{x}}$ { 1 } $\underset{x\to 0}{\lim }{(1+x^2)}^{\frac{1}{\tan x}}$ { 1 } $\underset{x\to +\mathrm{\infty }}{\lim }{x}^{\frac{x}{x^2+1}}$ { 1 } $\underset{x\to +\mathrm{\infty }}{\lim }{e}^{-x}{(e+\frac{2}{x})}^x$ { e^2/e|e^(2/e) } $\underset{x\to 0}{\lim }{(2-\cos x)}^{{\csc }^{2}x}$ { rad(e)|rad_2(e)|e^1/2|e^(1/2) } $\underset{x\to \mathrm{\infty }}{\lim }x(e^{\frac{1}{x}}-1)$ { 1 } $\underset{x\to \mathrm{\infty }}{\lim }{x}^2(1-\cos \frac{1}{x})$ { 1/2 } $\underset{x\to \frac{\pi }{4}}{\lim }\frac{\cos 2x}{\sin x-\cos x}$ { - rad(2)|- 2^1/2|- 2^(1/2) } $\underset{x\to 0}{\lim }\frac{e^{\sin 2x}-e^{\sin x}}{x}$ { 1 } $\underset{x\to 0^{+}}{\lim }{(1+{\sin }^{2}x)}^{\frac{1}{x^3}}$ { 1 } $\underset{x\to 0}{\lim }\frac{e^x-e^{-x}}{x}$ { 2 } $\underset{x\to -\mathrm{\infty }}{\lim }\frac{\sqrt{\frac{1+x}{x^3}}}{\sin \frac{1}{x}}$ { - 1 } $\underset{x\to 1}{\lim }(1-x)\tan \frac{x\pi }{2}$ { 2/pi } $\underset{x\to +0}{\lim }{({\tan }^2\sqrt{x}+1)}^{\frac{1}{2x}}$ { rad(e)| e^1/2| e^(1/2) } </quiz> Template:Avanzamento