limit as x approaches infinity of (1+sin(3/x))^{x+1}

Lösung

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

e^{3}

Dezimale

20.08553 \dots

Schritte zur Lösung

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

Vereinfache

(1 + sin (\frac{3}{x}))^{x + 1} : (1 + sin (\frac{3}{x}))^{x} (1 + sin (\frac{3}{x}))

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

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

mit Einschränkung des unbestimmten Ausdrucks

= x \to \infty lim ((1 + sin (\frac{3}{x}))^{x}) \cdot x \to \infty lim (1 + sin (\frac{3}{x}))

x \to \infty lim ((1 + sin (\frac{3}{x}))^{x}) = e^{3}

x \to \infty lim (1 + sin (\frac{3}{x})) = 1

= e^{3} \cdot 1

Vereinfache

= e^{3}