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Examples - Meira um l'Hopital

Dæmi 1: (Ath: $ a$ má vera $ \infty$ )
Látum $ n>0$ :

$\displaystyle \lim_{x\to \infty} \frac{\ln x}{x^n} = \lim_{x\to \infty} \frac{\frac{1}{x}}{nx^{n-1}} = \lim_{x\to \infty} \frac{1}{nx^n} = 0
$

þ.e.

$\displaystyle \lim_{x\to \infty} \frac{\ln x}{x^n} = 0 \qquad \forall n>0
$

M.ö.o. $ \ln x$ vex hægar en $ x^n$ fyrir hvaða $ n>0$ sem er, t.d. $ n = \frac{1}{100}$ .

Við getum átt við markgildi sem gefa $ 1^{\infty}, 0^0, \infty^{0}$ með því að taka logra.

Notum (ath. $ a$ má vera $ \infty$ )

$\displaystyle \lim_{x\to a} \ln f(x) = L \qquad \Rightarrow$    
$\displaystyle \lim_{x\to a} f(x) = \lim_{x\to a}e^{\ln f(x)} = e^{\lim_{x\to a} \ln f(x)} = e^L,$    

því $ e^x$ er samfellt fall.

Dæmi 2:

$\displaystyle \lim_{x\to 0} x^x \qquad (0^0)
$


$\displaystyle \lim_{x\to 0} \ln x^x$   $\displaystyle = \lim_{x\to 0} x\ln x$  
    $\displaystyle = \lim_{x\to 0} \frac{\ln x}{\frac{1}{x}} \qquad ($sem er$\displaystyle \quad \frac{-\infty}{\infty} )$  
    $\displaystyle = \lim_{x\to 0}\frac{\frac{1}{x}}{-\frac{1}{x^2}} = \lim_{x\to 0} \left( \frac{-x^2}{x} \right) = 0$  

$\displaystyle \Rightarrow \qquad \lim_{x\to 0} x^x = \lim_{x\to 0} e^{\ln x^x} = e^0 = 1
$



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