lim x→+∞ (√(5x^2+2x)+x√5) bằng

Question

\({\mathop{\lim }\limits_{x\to +\infty }} \left(\sqrt{5x^{2} +2x} +x\sqrt{5} \right)\) bằng

\(A. -\frac{\sqrt{5} }{5} . \)

\(B. 0. \)

\(C. 1.\)

\(D. +\infty .\)

Answer 1

Chọn D

Ta có \({\mathop{\lim }\limits_{x\to +\infty }} \left(\sqrt{5x^{2} +2x} +x\sqrt{5} \right)={\mathop{\lim }\limits_{x\to +\infty }} x\left(\sqrt{5+\frac{2}{x} } +\sqrt{5} \right).\)
\({\mathop{\lim }\limits_{x\to +\infty }} x=+\infty .\, \, \left(1\right) \)
\({\mathop{\lim }\limits_{x\to +\infty }} \left(\sqrt{5+\frac{2}{x} } +\sqrt{5} \right)=2\sqrt{5} .\, \, \, \left(2\right) \)

Từ \(\left(1\right),\, \, \left(2\right)\Rightarrow {\mathop{\lim }\limits_{x\to +\infty }} x\left(\sqrt{5+\frac{2}{x} } +\sqrt{5} \right)=+\infty .\)