Grenseverdisetningene

$$\begin{gathered} $ $ \begin{array}{rcl}\underset{x\to a}{\lim } \left(f\left(x\right)+g\left(x\right)\right)& =& \underset{x\to a}{\lim } f\left(x\right)+\underset{x\to a}{\lim } g\left(x\right)\\ & & \\ \underset{x\to a}{\lim } \left(f\left(x\right)-g\left(x\right)\right)& =& \underset{x\to a}{\lim } f\left(x\right)-\underset{x\to a}{\lim } g\left(x\right)\\ & & \\ \underset{x\to a}{\lim } \left(f\left(x\right)·g\left(x\right)\right)& =& \underset{x\to a}{\lim } f\left(x\right)·\underset{x\to a}{\lim } g\left(x\right)\\ & & \\ \underset{x\to a}{\lim } \frac{f\left(x\right)}{g\left(x\right)}& =& \frac{\underset{x\to a}{\lim } f\left(x\right)}{\underset{x\to a}{\lim } g\left(x\right)}\\ & & \mathrm{forutsatt} \mathrm{at} \underset{x\to a}{\lim } g\left(x\right)\ne 0\\ & & \end{array} \\ $$ \end{gathered}$$

$$ \begin{array}{rcl}\underset{x\to 1}{\lim }3x & =& \underset{x\to 1}{\lim }3·\underset{x\to 1}{\lim }x\\ & =& 3·1\\ & =& 3\end{array}$$

$$\begin{gathered} $ \begin{array}{rcl}\underset{x\to 2}{\lim }\left(x^2+3x\right) & =& \underset{x\to 2}{\lim }{x}^2+\underset{x\to 2}{\lim }3x\\ & = & 2^2+3·2\\ & = & 4+6 \\ & =& 10\end{array} \\ $ \end{gathered}$$

$$ \begin{array}{rcl}\underset{x\to 3}{\lim }\left(x^3+x^2-6^2\right) & =& \underset{x\to 3}{\lim }{x}^3+\underset{x\to 3}{\lim }{x}^2-\underset{x\to 3}{\lim }{6}^2\\ & =& 3^3+3^2-6^2 \\ & =& 27+9-36 \\ & =& 0\end{array}$$

$$ \begin{array}{rcl}\underset{x\to 3}{\lim }\frac{5x-10}{x^2} & =& 5·\frac{\underset{x\to 3}{\lim }\left(x-2\right)}{\underset{x\to 3}{\lim }{x}^2}\\ & & \\ & =& 5·\frac{3-2}{3^2} \\ & & \\ & =& 5·\frac{1}{9}\\ & & \\ & =& \frac{5}{9} \end{array}$$

$$ \begin{array}{rcl}\underset{x\to \infty }{\lim }\left(4x+3\right) & =& \underset{x\to \infty }{\lim }4x+\underset{x\to \infty }{\lim }3\end{array}$$

Siden $$ 4x$$ går mot uendelig når $$ x$$ går mot uendelig, eksisterer ikke grenseverdien.

$$ \begin{array}{rcl}\underset{x\to \infty }{\lim }\frac{3}{x}& = & \underset{x\to \infty }{\lim }3·\frac{1}{x}\\ & & \\ & =& 3·\underset{x\to \infty }{\lim }\frac{1}{x}\\ & & \\ & =& 3·0\\ & & \\ & =& 0\end{array}$$