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Examples - Dæmi

$\displaystyle f(x) = \begin{cases}
-x + 1 \qquad &x < 0\\
0 \qquad &x = 0\\
\frac{1}{2} \qquad &x > 0
\end{cases}$

$\displaystyle \lim_{x\to 0^{-}} f(x) = 1 \qquad \lim_{x\to 0^{+}} f(x) = \frac{1}{2}$    
en$\displaystyle \qquad f(0) = 0.$    

$\displaystyle f(x) = \sqrt{\frac{x^2 + 3x -1}{x^3 + 4x^2 + 1}}
$


$\displaystyle \lim_{x\to 1}f(x)$   $\displaystyle = \sqrt{\lim_{x\to 1} \left( \frac{x^2 + 3x -1}{x^3 + 4x^2 + 1}\right)}$  
    $\displaystyle = \sqrt{\frac{\lim_{x\to 1}(x^2 + 3x - 1)}{\lim_{x\to 1} (x^3 + 4x^2 + 1)}}$  
    $\displaystyle = \sqrt{\frac{\lim_{x\to 1} x^2 + \lim_{x\to 1} 3x + \lim_{x\to 1}(-1)}{\lim_{x\to 1}x^3 + \lim_{x\to 1}(4x^2) + \lim_{x\to 1} 1}}$  
    $\displaystyle = \sqrt{\frac{\left( \lim_{x\to 1} x\right)^2 + 3\lim_{x\to 1}x - 1}{\left(\lim_{x\to1} x\right)^3 + 4 \left( \lim_{x\to 1} x \right)^2 + 1}}$  
    $\displaystyle = \sqrt{\frac{1^2 + 3\cdot 1 -1}{1^3 + 4\cdot 1^2 + 1}}$  
    $\displaystyle = \sqrt{\frac{3}{6}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$  



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