limit as x approaches 0+of (ln(x))/(csc(x))

Lösung

x \to 0 + lim (\frac{ln ( x )}{csc ( x )})

0

Schritte zur Lösung

x \to 0 + lim (\frac{ln ( x )}{csc ( x )})

Verwende die grundlegende trigonometrische Identität:

\frac{1}{csc ( x )} = sin (x)

= x \to 0 + lim (ln (x) sin (x))

Umformung für L'Hopital

= x \to 0 + lim (\frac{ln ( x )}{\frac{1}{s i n ( x )}})

Wende das L'Hopital Theorem an

= x \to 0 + lim (\frac{\frac{1}{x}}{- cot ( x ) csc ( x )})

Vereinfache

\frac{\frac{1}{x}}{- cot ( x ) csc ( x )} : - \frac{1}{x cot ( x ) csc ( x )}

= x \to 0 + lim (- \frac{1}{x cot ( x ) csc ( x )})

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

= - x \to 0 + lim (\frac{1}{x cot ( x ) csc ( x )})

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

= - \frac{lim _{x \to 0 +} ( 1 )}{lim _{x \to 0 +} ( x cot ( x ) csc ( x ) )}

x \to 0 + lim (1) = 1

x \to 0 + lim (x cot (x) csc (x)) = \infty

= - \frac{1}{\infty}

Wende die Unendlichkeitseigenschaft an:

\frac{c}{\infty} = 0

= 0

ma c l a u r in f (x) = \frac{1}{( 1 - x ) ^{2}}

x \to 0 + lim (2 (tan (x))^{x})

\frac{d y}{d x} = \frac{( x ^{2} + y ^{2} )}{2 x y}

y^{^{''}} - y = \frac{e ^{x}}{e ^{x} + 1}

\frac{d}{d x} (ln (\frac{x + 1}{x - 2}))