Question

Предел Lim x->∞ sqrt(x^2+5x)-x

Answer 1

Ответ:

2.5

Пошаговое объяснение:

Lim x->∞ sqrt(x²+5x)-x

Lim x->∞ (√(x²+5x)-x)

Lim x->∞ 5x/(x+√(x²+5x))

5 Lim x->∞  x/(x+√(x²+5x))

5 Lim x->∞ 1/(√x²+5x/x + 1)

5 1/ (Lim x->∞ (√x²+5x/x + 1))

5/(Lim x->∞ √x²+5x/x² + 1)

5/(√(Lim x->∞ x²+5x/x² + 1))

5/(√(Lim x->∞ 1+5/x/1 + 1)

5/√1 + 1

5/2

2.5

Answer 2

$\displaystyle \lim_{x \to \infty}\left(\sqrt{x^2+5x}-x\right)=\lim_{x \to \infty}\frac{\Big(\sqrt{x^2+5x}-x\Big)\Big(\sqrt{x^2+5x}+x\Big)}{\sqrt{x^2+5x}+x}=\\ \\ \\ =\lim_{x \to \infty}\frac{5x}{\sqrt{x^2+5x}+x}=\lim_{x \to \infty}\frac{5}{\sqrt{1+\frac{5}{x}}+1}=\frac{5}{2}.$