limit as x approaches 0 of (3x^2)/(1-cos^2(x/2))

Lösung

x \to 0 lim (\frac{3 x ^{2}}{1 - cos ^{2} ( \frac{x}{2} )})

12

Schritte zur Lösung

x \to 0 lim (\frac{3 x ^{2}}{1 - cos ^{2} ( \frac{x}{2} )})

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

= 3 \cdot x \to 0 lim (\frac{x ^{2}}{1 - cos ^{2} ( \frac{x}{2} )})

Wende das L'Hopital Theorem an

= 3 \cdot x \to 0 lim (\frac{2 x}{\frac{s i n ( x )}{2}})

Vereinfache

\frac{2 x}{\frac{s i n ( x )}{2}} : 4 x csc (x)

= 3 \cdot x \to 0 lim (4 x csc (x))

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

= 3 \cdot 4 \cdot x \to 0 lim (x csc (x))

Umformung für L'Hopital

= 3 \cdot 4 \cdot x \to 0 lim (\frac{x}{\frac{1}{c s c ( x )}})

Wende das L'Hopital Theorem an

= 3 \cdot 4 \cdot x \to 0 lim (\frac{1}{cos ( x )})

Setze den Wert

x = 0

ein

= 3 \cdot 4 \cdot \frac{1}{cos ( 0 )}

Vereinfache

3 \cdot 4 \cdot \frac{1}{cos ( 0 )} : 12

= 12

\int \frac{x ^{3} + x ^{2} + 2}{x ^{2} - x} d x

\int \frac{1}{( x ^{2} + 2 x + 3 ) ^{\frac{3}{2}}} d x

\frac{d y}{d x} = 6 y x

x \to 0 lim (\frac{5}{x})

\frac{d y}{d x} + 5 e^{x + y} = 0