limit as x approaches 0 of (pi/x}{cot(\frac{pix)/2)}

Lösung

x \to 0 lim (\frac{\frac{π}{x}}{cot ( \frac{π x}{2} )})

\frac{π ^{2}}{2}

Dezimale

4.93480 \dots

Schritte zur Lösung

x \to 0 lim (\frac{\frac{π}{x}}{cot ( \frac{π x}{2} )})

Wende Bruchregel an:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= x \to 0 lim (\frac{π}{x cot ( \frac{π x}{2} )})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= π \cdot x \to 0 lim (\frac{1}{x cot ( \frac{π x}{2} )})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

mit Einschränkung des unbestimmten Ausdrucks

= π \frac{lim _{x \to 0} ( 1 )}{lim _{x \to 0} ( x cot ( \frac{π x}{2} ) )}

x \to 0 lim (1) = 1

x \to 0 lim (x cot (\frac{π x}{2})) = \frac{2}{π}

= π \frac{1}{\frac{2}{π}}

Vereinfache

π \frac{1}{\frac{2}{π}} : \frac{π ^{2}}{2}

= \frac{π ^{2}}{2}

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