limit as x approaches-infinity of (sqrt(ax^2+bx+c))/(d-ax)

Lösung

x \to - \infty lim (\frac{a x ^{2} + b x + c}{d - a x})

\frac{1}{a}

Schritte zur Lösung

x \to - \infty lim (\frac{a x ^{2} + b x + c}{d - a x})

Teile durch den größten gemeinsamen Potenznenner:

\frac{- a + \frac{b}{x} + \frac{c}{x ^{2}}}{\frac{d}{x} - a}

= x \to - \infty lim \frac{- a + \frac{b}{x} + \frac{c}{x ^{2}}}{\frac{d}{x} - a}

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

= - x \to - \infty lim \frac{a + \frac{b}{x} + \frac{c}{x ^{2}}}{\frac{d}{x} - a}

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 - \infty} ( a + \frac{b}{x} + \frac{c}{x ^{2}} )}{lim _{x \to - \infty} ( \frac{d}{x} - a )}

x \to - \infty lim (a + \frac{b}{x} + \frac{c}{x ^{2}}) = a

x \to - \infty lim (\frac{d}{x} - a) = - a

= - \frac{a}{- a}

Vereinfache

- \frac{a}{- a} : \frac{1}{a}

= \frac{1}{a}

x \to 0 lim (4 x^{2} + 3 x - 2)

x \to 0 lim ((\frac{1 + 1}{x})^{x})

x \to 0 + lim (\frac{x}{e ^{- \frac{8}{x}}})

x \to - \infty lim (x, - 1)

x \to - \infty lim (\frac{6 x}{x + cos ( x )})