Esercizi di matematica per le superiori/Limiti

Richiami di teoria: Limiti.

1

Seleziona i limiti al finito corretti e, dopo averli corretti, trascrivili su un foglio e dimostra che questi limiti sono corretti:

	$\underset{x\to -3}{lim}\frac{x+3}{1+x^2}=0$
	$\underset{x\to 8}{lim}{\log }_2(x+8)=4$
	$\underset{x\to -1}{lim}\frac{2x^2+5x+3}{4x^2-3x+2}=3$
	$\underset{x\to 1}{lim}(x^2+x+8)=10$
	$\underset{x\to \frac{\pi }{3}}{lim}\frac{\sin 2x+\cos x}{\sin 5x-\cos (2x-\frac{\pi }{3})}=1$
	$\underset{x\to 3}{lim}(x^3+x+1)=30$
	$\underset{x\to 0}{lim}\frac{e^x}{2^x}=1$
	$\underset{x\to \pi }{lim}\frac{\tan x+\cos x}{\cos x+\sin 2x}=0$
	$\underset{x\to 2}{lim}\frac{x^2+4}{2x-4}=2$
	$\underset{x\to \frac{1}{2}}{lim}\frac{2x^3+x^2-2x-1}{4x^2-1}+\frac{5x}{4}+x=1$

2

Calcola i seguenti limiti al finito di queste funzioni:

$\underset{x\to 1}{lim}-\frac{x+3}{x-2}$

$\underset{x\to 0}{lim}{\log }_2(x+8)$

$\underset{x\to -1}{lim}\frac{x^2-3x+2}{x^2-7x+12}-\frac{7x}{10}$

$\underset{x\to 0}{lim}{\log }_2(8^x+x^8+8)$

$\underset{x\to \frac{\pi }{2}}{lim}\frac{\sin x+\cos x}{\sin 2x-\cos (x-\frac{\pi }{2})}$

3

Calcola i seguenti limiti all'infinito di queste funzioni:

$\underset{x\to +\mathrm{\infty }}{lim}(-x+3)$

$\underset{x\to +\mathrm{\infty }}{lim}{\log }_2(x+8)$

$\underset{x\to -\mathrm{\infty }}{lim}\frac{x^2-3x+2}{x^2-7x+12}-\frac{7x}{10}$

$\underset{x\to \mathrm{\infty }}{lim}\frac{3x^5-3x+2}{12-x^2+3x^5+x+x^3}$

4

Calcola i seguenti limiti al finito di queste funzioni:

$\underset{x\to 1}{lim}\frac{x^3-x^2-x+1}{x^3+x^2-x-1}$

$\underset{x\to \mathrm{\infty }}{lim}\frac{x^3+x}{x^4-3x^2+1}$

$\underset{x\to 1}{lim}(\frac{1}{1-x}-\frac{3}{1-x^3})$

$\underset{x\to 1}{lim}\frac{x^2-\sqrt{x}}{\sqrt{x}-1}$

$\underset{x\to +\mathrm{\infty }}{lim}\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}$

$\underset{x\to 0}{lim}\frac{\sqrt{1+x+x^2}-\sqrt{1-x+x^2}}{x^2-x}$

$\underset{x\to 1}{lim}\frac{\sqrt{x+3}-\sqrt{3x+1}}{x-1}$

$\underset{x\to 0}{lim}\frac{\sqrt{1+x^2}-1}{x^2}$

$\underset{x\to 0}{lim}\frac{\sin x}{|x|}$

$\underset{x\to 0}{lim}\frac{\sqrt{1+x\sin x}-1}{x^2}$

$\underset{x\to \frac{\pi }{4}}{lim}[(1-\tan x)\tan 2x]$

$\underset{x\to 0}{lim}\frac{{\sin }^{2}x}{1-\cos x^3}$

$\underset{x\to 0}{lim}\frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}$

$\underset{x\to 0}{lim}\frac{\sin 2x}{\ln (1+x)}$

$\underset{x\to 0}{lim}\frac{\sin 3x-\sin x}{\ln (1+x)}$

$\underset{x\to 0}{lim}\frac{\ln (x+2)-\ln 2}{x}$

$\underset{x\to 0}{lim}\frac{\ln (\cos x)}{x^2}$

$\underset{x\to \mathrm{\infty }}{lim}{\left(\frac{x^2+2}{x^2}\right)}^{x^2+2}$

$\underset{x\to 1}{lim}{\left(\frac{x-1}{x^2-1}\right)}^{x+1}$

$\underset{x\to \mathrm{\infty }}{lim}{\left(\frac{x^2+2}{2x^2+1}\right)}^{x^2}$

$\underset{x\to 0}{lim}{(\cos x)}^{\frac{1}{x}}$

$\underset{x\to 0}{lim}{(1+x^2)}^{\frac{1}{\tan x}}$

$\underset{x\to +\mathrm{\infty }}{lim}{x}^{\frac{x}{x^2+1}}$

$\underset{x\to +\mathrm{\infty }}{lim}{e}^{-x}{(e+\frac{2}{x})}^x$

$\underset{x\to 0}{lim}{(2-\cos x)}^{{\csc }^{2}x}$

$\underset{x\to \mathrm{\infty }}{lim}x(e^{\frac{1}{x}}-1)$

$\underset{x\to \mathrm{\infty }}{lim}{x}^2(1-\cos \frac{1}{x})$

$\underset{x\to \frac{\pi }{4}}{lim}\frac{\cos 2x}{\sin x-\cos x}$

$\underset{x\to 0}{lim}\frac{e^{\sin 2x}-e^{\sin x}}{x}$

$\underset{x\to 0^{+}}{lim}{(1+{\sin }^{2}x)}^{\frac{1}{x^3}}$

$\underset{x\to 0}{lim}\frac{e^x-e^{-x}}{x}$

$\underset{x\to -\mathrm{\infty }}{lim}\frac{\sqrt{\frac{1+x}{x^3}}}{\sin \frac{1}{x}}$

$\underset{x\to 1}{lim}(1-x)\tan \frac{x\pi }{2}$

$\underset{x\to +0}{lim}{({\tan }^2\sqrt{x}+1)}^{\frac{1}{2x}}$