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You are here: Home Test Department Heildunarreglur Regla l'Hopital l'Hopital 1

Details - l'Hopital 1

Skoðum nú hvernig finna má markgildi

$\displaystyle \lim_{x\to a} \frac{f(x)}{g(x)}
$

þar sem $ f(a) = g(a) = 0$ .
þ.e. ,, $ \frac{0}{0}$ markgildi``.

Setning:     (l'Hopital 1) $ f(a) = g(a) = 0$ ;     $ f'(a)$ og $ g'(a)$ eru til og $ g'(a) \neq 0$ . Þá er

$\displaystyle \lim_{x\to a}\frac{f(x)}{g(x)} = \frac{f'(a)}{g'(a)}
$

Sönnun:
$\displaystyle \lim_{x\to a} \frac{f(x)}{g(x)}
$   $\displaystyle = \lim_{x\to a}\frac{f(x) - f(a)}{g(x) - g(a)}$  
    $\displaystyle = \lim_{x\to a} \frac{\left( \frac{f(x) - f(a)}{x-a}\right)}{\lef...
...{f(x) - f(a)}{x-a}\right)}{\lim_{x\to a} \left( \frac{g(x) - g(a)}{x-a}\right)}$  
    $\displaystyle = \frac{f'(a)}{g'(a)}$  



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