limit as x approaches 0+of 6(1/x-cot(x))

Lösung

x \to 0 + lim (6 (\frac{1}{x} - cot (x)))

0

Schritte zur Lösung

x \to 0 + lim (6 (\frac{1}{x} - cot (x)))

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

= 6 \cdot x \to 0 + lim (\frac{1}{x} - cot (x))

Multipliziere mit der Konjugation von

\frac{1}{x} - cot (x) : \frac{\frac{1}{x ^{2}} - cot ^{2} ( x )}{\frac{1}{x} + cot ( x )}

= 6 \cdot x \to 0 + lim (\frac{\frac{1}{x ^{2}} - cot ^{2} ( x )}{\frac{1}{x} + cot ( 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

= 6 \cdot \frac{lim _{x \to 0 +} ( \frac{1}{x ^{2}} - cot ^{2} ( x ) )}{lim _{x \to 0 +} ( \frac{1}{x} + cot ( x ) )}

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

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

= 6 \cdot \frac{\frac{2}{3}}{\infty}

Vereinfache

6 \cdot \frac{\frac{2}{3}}{\infty} : 0

= 0

x \to 0 lim (x^{4} \cdot sin (\frac{1}{x}))

x \to 0 lim (\frac{1}{x \cdot csc ( x )})

x \to - 3 lim (ln (4 x + 14))

x \to 0 lim (\frac{sin ( 3 x ) - 1}{x})

x \to 0 lim (\frac{x}{[ x ]})