Exercicis de límits. Límits immediats i quasi immediats de funcions

Exercici 1

Calcula els següents límits, indicant que passa quan \(x\rightarrow +\infty\), \(x\rightarrow -\infty\) i \(x\rightarrow \infty\):

a) \(\displaystyle\lim_{x\rightarrow \infty} x^4 \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} x^4 = +\infty\)	\(\displaystyle\lim_{x\rightarrow -\infty} x^4 = +\infty\)	\(\displaystyle\lim_{x\rightarrow \infty} x^4 = +\infty\)
b) \(\displaystyle\lim_{x\rightarrow \infty} x^5 \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} x^5 = +\infty\)	\(\displaystyle\lim_{x\rightarrow -\infty} x^5 = -\infty\)	\(\displaystyle\lim_{x\rightarrow \infty} x^5 = \infty\)
c) \(\displaystyle\lim_{x\rightarrow \infty} \frac{1}{x^3} \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} \frac{1}{x^3} = 0\)	\(\displaystyle\lim_{x\rightarrow -\infty} \frac{1}{x^3} = 0\)	\(\displaystyle\lim_{x\rightarrow \infty} \frac{1}{x^3} = 0\)
d) \(\displaystyle\lim_{x\rightarrow \infty} \frac{1}{x^6} \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} \frac{1}{x^6} = 0\)	\(\displaystyle\lim_{x\rightarrow -\infty} \frac{1}{x^6} = 0\)	\(\displaystyle\lim_{x\rightarrow \infty} \frac{1}{x^6} = 0\)
e) \(\displaystyle\lim_{x\rightarrow \infty} \sqrt[5]{x} \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} \sqrt[5]{x} = +\infty\)	\(\displaystyle\lim_{x\rightarrow -\infty} \sqrt[5]{x} = -\infty\)	\(\displaystyle\lim_{x\rightarrow \infty} \sqrt[5]{x} = \infty\)
f) \(\displaystyle\lim_{x\rightarrow \infty} \sqrt[4]{x} \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} \sqrt[4]{x} = +\infty\)	\(\displaystyle\nexists\lim_{x\rightarrow -\infty} \sqrt[4]{x} \)	\(\displaystyle\nexists\lim_{x\rightarrow \infty} \sqrt[4]{x} \)

Exercici 2

Calcula els següents límits, indicant que passa quan \(x\rightarrow +\infty\), \(x\rightarrow -\infty\) i \(x\rightarrow \infty\):

a) \(\displaystyle\lim_{x\rightarrow \infty} 3^x \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} 3^x = +\infty\)	\(\displaystyle\lim_{x\rightarrow -\infty} 3^x = 0\)	\(\displaystyle\nexists\lim_{x\rightarrow \infty} 3^x\)
b) \(\displaystyle\lim_{x\rightarrow \infty} -3^x \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} -3^x = -\infty\)	\(\displaystyle\lim_{x\rightarrow -\infty} -3^x = 0\)	\(\displaystyle\nexists\lim_{x\rightarrow \infty} -3^x\)
c) \(\displaystyle\lim_{x\rightarrow \infty} \text{0,5}^x \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} \text{0,5}^x = 0\)	\(\displaystyle\lim_{x\rightarrow -\infty} \text{0,5}^x = +\infty\)	\(\displaystyle\nexists\lim_{x\rightarrow \infty} \text{0,5}^x \)
d) \(\displaystyle\lim_{x\rightarrow \infty} -\text{0,5}^x \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} -\text{0,5}^x = 0\)	\(\displaystyle\lim_{x\rightarrow -\infty} -\text{0,5}^x = -\infty\)	\(\displaystyle\nexists\lim_{x\rightarrow \infty} -\text{0,5}^x \)
e) \(\displaystyle\lim_{x\rightarrow \infty} 5^{-x} \)	Solució:	\(\displaystyle\lim_{x\rightarrow +\infty} 5^{-x} = 0\)	\(\displaystyle\lim_{x\rightarrow -\infty} 5^{-x} = +\infty\)	\(\displaystyle\nexists\lim_{x\rightarrow \infty} 5^{-x}\)

Exercici 3

Calcula els següents límits:

a) \(\displaystyle\lim_{x\rightarrow 0}\left(\log{x}\right)\)	Solució:	\(\displaystyle\nexists\lim_{x\rightarrow 0^-}\left(\log{x}\right)\)	\(\displaystyle\lim_{x\rightarrow 0^+}\left(\log{x}\right)= -\infty\)	\(\displaystyle\nexists\lim_{x\rightarrow 0}\left(\log{x}\right)\)
b) \(\displaystyle\lim_{x\rightarrow 1}\left(\log{x}\right)\)	Solució:			\(\displaystyle\lim_{x\rightarrow 1}\left(\log{x}\right)=0\)
c) \(\displaystyle\lim_{x\rightarrow \infty}\left(\log{x}\right)\)	Solució:	\(\displaystyle\nexists\lim_{x\rightarrow -\infty}\left(\log{x}\right)\)	\(\displaystyle\lim_{x\rightarrow +\infty}\left(\log{x}\right)=+\infty\)	\(\displaystyle\nexists\lim_{x\rightarrow \infty}\left(\log{x}\right)\)
d) \(\displaystyle\lim_{x\rightarrow 0}\left(\log_\text{0,1}{x}\right)\)	Solució:	\(\displaystyle\nexists\lim_{x\rightarrow 0^-}\left(\log_\text{0,1}{x}\right)\)	\(\displaystyle\lim_{x\rightarrow 0^+}\left(\log_\text{0,1}{x}\right)=+\infty\)	\(\displaystyle\nexists\lim_{x\rightarrow 0}\left(\log_\text{0,1}{x}\right)\)
e) \(\displaystyle\lim_{x\rightarrow \infty}\left(\log_\text{0,1}{x}\right)\)	Solució:	\(\displaystyle\nexists\lim_{x\rightarrow -\infty}\left(\log_\text{0,1}{x}\right)\)	\(\displaystyle\lim_{x\rightarrow +\infty}\left(\log_\text{0,1}{x}\right)=-\infty\)	\(\displaystyle\nexists\lim_{x\rightarrow \infty}\left(\log_\text{0,1}{x}\right)\)

Exercici 4

Calcula els següents límits:

a) \(\displaystyle\lim_{x\rightarrow -\infty}\left( 3x^4-2x^5 \right)\)	Solució:	\(\displaystyle \lim_{x\rightarrow -\infty}\left( 3x^4-2x^5 \right) = \lim_{x\rightarrow -\infty}\left( -2x^5 \right) = +\infty \)
b) \(\displaystyle\lim_{x\rightarrow\pm\infty}\left(2x^2-3\sqrt{x^5}\right)\)	Solució:	\(\displaystyle \lim_{x\rightarrow \pm\infty}\left( 2x^2-3\sqrt{x^5} \right) = \lim_{x\rightarrow \pm\infty}\left( 2x^2-3x^\frac{5}{2} \right) = \lim_{x\rightarrow \pm\infty}\left( -3x^\frac{5}{2} \right) = \mp\infty \)
c) \(\displaystyle\lim_{x\rightarrow +\infty}\left(3^x-x^4\right)\)	Solució:	\(\displaystyle \lim_{x\rightarrow +\infty}\left(3^x-x^4\right) = \lim_{x\rightarrow +\infty} {3^x} = +\infty \)
d) \(\displaystyle\lim_{x\rightarrow -\infty}\left(3^x-x^4\right)\)	Solució:	\(\displaystyle \lim_{x\rightarrow -\infty}\left(3^x-x^4\right) = \lim_{x\rightarrow -\infty} {-x^4} = -\infty \)
e) \(\displaystyle\lim_{x\rightarrow +\infty}\left(x-\log_2{x}\right)\)	Solució:	\(\displaystyle \lim_{x\rightarrow +\infty}\left(x-\log_2{x}\right) = \lim_{x\rightarrow +\infty} {x} = +\infty \)
f) \(\displaystyle\lim_{x\rightarrow -\infty}\left(\log{x}-\log_2{x}\right)\)	Solució:	\(\displaystyle\nexists\lim_{x\rightarrow -\infty}\left(\log{x}-\log_2{x}\right) \)
g) \(\displaystyle\lim_{x\rightarrow +\infty}\left(4x^3-5^x+\log_5{x}\right)\)	Solució:	\(\displaystyle \lim_{x\rightarrow +\infty}\left(4x^3-5^x+\log_5{x}\right) = \lim_{x\rightarrow +\infty}\left(-5^x\right) = -\infty \)

Exercici 5

Calcula els següents límits:

a) \(\displaystyle\lim_{x\rightarrow \infty}\frac{2x^2+3x^3}{4x^3-5x}\)	Solució:	\(\displaystyle \lim_{x\rightarrow \pm\infty}\frac{2x^2+3x^3}{4x^3-5x} = \frac{3}{4}\)
b) \(\displaystyle\lim_{x\rightarrow \infty}\frac{4x^4-5x^2}{-2x^3+x^2-6x}\)	Solució:	\(\displaystyle \lim_{x\rightarrow \pm\infty}\frac{4x^4-5x^2}{-2x^3+x^2-6x} = \mp\infty \)
c) \(\displaystyle\lim_{x\rightarrow \infty}\frac{-5x^3+3x^2}{-4x^4+6x^2-1}\)	Solució:	\(\displaystyle \lim_{x\rightarrow \infty}\frac{-5x^3+3x^2}{-4x^4+6x^2-1} = 0 \)
d) \(\displaystyle\lim_{x\rightarrow +\infty}\frac{x^3+e^x}{x^4+\log_2{x}}\)	Solució:	\( \displaystyle \lim_{x\rightarrow +\infty}\frac{x^3+e^x}{x^4+\log_2{x}} = \lim_{x\rightarrow +\infty}\frac{e^x}{x^4} = +\infty \)
e) \(\displaystyle\lim_{x\rightarrow -\infty}\frac{x^3+e^x}{x^4+\log_2{x}}\)	Solució:	\( \displaystyle \nexists\lim_{x\rightarrow -\infty}{\log_2{x}} \Rightarrow \nexists\lim_{x\rightarrow -\infty}\frac{x^3+e^x}{x^4+\log_2{x}} \)
f) \(\displaystyle\lim_{x\rightarrow +\infty}\frac{3^x-e^x}{2^x-e^x}\)	Solució:	\( \displaystyle \lim_{x\rightarrow +\infty}{\frac{3^x-e^x}{2^x-e^x}} = \lim_{x\rightarrow +\infty}{\frac{3^x}{-e^x}} = -\infty \)