limit as x approaches 0 of (x-arctan(x))/(x^2tan(x))

Lösung

x \to 0 lim (\frac{x - arctan ( x )}{x ^{2} tan ( x )})

\frac{1}{3}

Dezimale

0.33333 \dots

Schritte zur Lösung

x \to 0 lim (\frac{x - arctan ( x )}{x ^{2} tan ( x )})

Wende das L'Hopital Theorem an

= x \to 0 lim (\frac{1 - \frac{1}{x ^{2} + 1}}{2 x tan ( x ) + sec ^{2} ( x ) x ^{2}})

Wende das L'Hopital Theorem an

= x \to 0 lim (\frac{\frac{2 x}{( x ^{2} + 1 ) ^{2}}}{2 x ^{2} sec ^{2} ( x ) tan ( x ) + 4 x sec ^{2} ( x ) + 2 tan ( x )})

Vereinfache

\frac{\frac{2 x}{( x ^{2} + 1 ) ^{2}}}{2 x ^{2} sec ^{2} ( x ) tan ( x ) + 4 x sec ^{2} ( x ) + 2 tan ( x )} : \frac{x}{( x ^{2} + 1 ) ^{2} ( x ^{2} sec ^{2} ( x ) tan ( x ) + 2 x sec ^{2} ( x ) + tan ( x ) )}

= x \to 0 lim (\frac{x}{( x ^{2} + 1 ) ^{2} ( x ^{2} sec ^{2} ( x ) tan ( x ) + 2 x sec ^{2} ( x ) + tan ( x ) )})

Wende das L'Hopital Theorem an

= x \to 0 lim (\frac{1}{- 2 x ^{6} sec ^{2} ( x ) + 3 x ^{6} sec ^{4} ( x ) + 10 x ^{5} sec ^{2} ( x ) tan ( x ) + 7 x ^{4} sec ^{2} ( x ) + 6 x ^{4} sec ^{4} ( x ) + 4 x ^{3} tan ( x ) + 16 x ^{3} sec ^{2} ( x ) tan ( x ) + 12 x ^{2} sec ^{2} ( x ) + 3 x ^{2} sec ^{4} ( x ) + 4 x tan ( x ) + 6 x sec ^{2} ( x ) tan ( x ) + 3 sec ^{2} ( x )})

Setze den Wert

x = 0

ein

= \frac{1}{- 2 \cdot 0 ^{6} sec ^{2} ( 0 ) + 3 \cdot 0 ^{6} sec ^{4} ( 0 ) + 10 \cdot 0 ^{5} sec ^{2} ( 0 ) tan ( 0 ) + 7 \cdot 0 ^{4} sec ^{2} ( 0 ) + 6 \cdot 0 ^{4} sec ^{4} ( 0 ) + 4 \cdot 0 ^{3} tan ( 0 ) + 16 \cdot 0 ^{3} sec ^{2} ( 0 ) tan ( 0 ) + 12 \cdot 0 ^{2} sec ^{2} ( 0 ) + 3 \cdot 0 ^{2} sec ^{4} ( 0 ) + 4 \cdot 0 \cdot tan ( 0 ) + 6 \cdot 0 \cdot sec ^{2} ( 0 ) tan ( 0 ) + 3 sec ^{2} ( 0 )}

Vereinfache

\frac{1}{- 2 \cdot 0 ^{6} sec ^{2} ( 0 ) + 3 \cdot 0 ^{6} sec ^{4} ( 0 ) + 10 \cdot 0 ^{5} sec ^{2} ( 0 ) tan ( 0 ) + 7 \cdot 0 ^{4} sec ^{2} ( 0 ) + 6 \cdot 0 ^{4} sec ^{4} ( 0 ) + 4 \cdot 0 ^{3} tan ( 0 ) + 16 \cdot 0 ^{3} sec ^{2} ( 0 ) tan ( 0 ) + 12 \cdot 0 ^{2} sec ^{2} ( 0 ) + 3 \cdot 0 ^{2} sec ^{4} ( 0 ) + 4 \cdot 0 \cdot tan ( 0 ) + 6 \cdot 0 \cdot sec ^{2} ( 0 ) tan ( 0 ) + 3 sec ^{2} ( 0 )} : \frac{1}{3}

= \frac{1}{3}

x \to a lim (f (x) + h (x))

x \to 5 lim (2 x - 1)

x \to 2 - lim (\frac{( x + 2 )}{x - 2})

x \to \infty lim (\frac{2}{x} + 4)

x \to - 8 lim (5)