Question

Hasil dari [tex]\displaystyle \lim_{x\to \infty}\left ( \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x}}}}-\sqrt{x} \right )[/tex] yaitu ...

Answer 1

Verified answer

Penjelasan dengan langkah-langkah:

Syarat

∞ = bernilai positif

[tex]\displaystyle \lim_{x\to \infty}\left ( \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x}}}}-\sqrt{x} \right ) \\ \\ [/tex]

Rasionalkan

[tex] = \frac{( \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x} } } } - \sqrt{x}) \times ( \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x} } } } + \sqrt{x}) }{( \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x} } } } + \sqrt{x})} \\ \\ (x + y)(x - y) = {x}^{2} - {y}^{2} \\ \\ = \frac{\cancel {x} + \sqrt{x + \sqrt{x + \sqrt{x} } } -\cancel{ x}}{ \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x} } } } + \sqrt{x}} \\ \\ = \frac{ \sqrt{x + \sqrt{x + \sqrt{x} } } }{ \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x} } } } + \sqrt{x}} \\ \\ = \frac{ \sqrt{x} ( \sqrt{1 + \sqrt{ \frac{1}{x} + \sqrt{ \frac{1}{ {x}^{3} } } } }) }{\sqrt{x} ( \sqrt{1 + \sqrt{ \frac{1}{x} + \sqrt{ \frac{1}{ {x}^{3} } + \sqrt{ \frac{1}{ {x}^{7} } } } } + 1})} \\ \\ = \frac{ ( \sqrt{1 + \sqrt{ \frac{1}{x} + \sqrt{ \frac{1}{ {x}^{3} } } } }) }{ ( \sqrt{1 + \sqrt{ \frac{1}{x} + \sqrt{ \frac{1}{ {x}^{3} } + \sqrt{ \frac{1}{ {x}^{7} } } } } + 1})} \\ \\subs \\ \\ = \frac{ \sqrt{1 + \sqrt{0 + \sqrt{0} } } }{ \sqrt{1 + \sqrt{0 + \sqrt{0 + \sqrt{0} } } } + 1 } \\ \\ = \frac{1}{2} [/tex]