limit as x approaches infinity of (x^3+2x^2+3x+4)/(4x^2+3x^2+2x+1)

Lösung

x \to \infty lim (\frac{x ^{3} + 2 x ^{2} + 3 x + 4}{4 x ^{2} + 3 x ^{2} + 2 x + 1})

\infty

Schritte zur Lösung

x \to \infty lim (\frac{x ^{3} + 2 x ^{2} + 3 x + 4}{4 x ^{2} + 3 x ^{2} + 2 x + 1})

Teile durch den größten gemeinsamen Potenznenner:

\frac{x + 2 + \frac{3}{x} + \frac{4}{x ^{2}}}{4 + 3 + \frac{2}{x} + \frac{1}{x ^{2}}}

= x \to \infty lim (\frac{x + 2 + \frac{3}{x} + \frac{4}{x ^{2}}}{4 + 3 + \frac{2}{x} + \frac{1}{x ^{2}}})

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} ( x + 2 + \frac{3}{x} + \frac{4}{x ^{2}} )}{lim _{x \to \infty} ( 4 + 3 + \frac{2}{x} + \frac{1}{x ^{2}} )}

x \to \infty lim (x + 2 + \frac{3}{x} + \frac{4}{x ^{2}}) = \infty

x \to \infty lim (4 + 3 + \frac{2}{x} + \frac{1}{x ^{2}}) = 7

= \frac{\infty}{7}

Wende die Unendlichkeitseigenschaft an:

\frac{\infty}{c} = \infty

= \infty

x \to - \infty lim (5 - x^{2})

x \to 2 + lim (x (2))

x \to 3 + lim (x - 3)

x \to \infty lim (\frac{x ^{2} - 6}{2 + x - 3 x ^{2}})

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