Jawab:
6.
[tex]= \lim_{x \to \infty} \sqrt{\frac{1+x}{1-x} } \times\sqrt{\frac{1+x}{1+x} }[/tex]
[tex]= \lim_{x \to \infty} \frac{1+x}{\sqrt{1-x^{2} } }[/tex] ⇒ dikalikan [tex]\frac{\frac{1}{x} }{\frac{1}{x} }[/tex]
[tex]= \lim_{x \to \infty} \frac{\frac{1}{x}+\frac{x}{x} }{\sqrt{\frac{1}{x^{2} } -\frac{x^{2} }{x^{2} } } }[/tex]
[tex]= \frac{0+1}{\sqrt{0-1} }[/tex]
=[tex]\frac{1}{\sqrt{-1} }[/tex]
(hasilnya merupakan bilang kompleks)
7.
[tex]= \lim_{x \to \infty} \frac{4x^{2} +20x+25}{9x^{2} -6x+1} \ (masih\ \frac{\infty}{\infty} )[/tex]
[tex]= \lim_{x \to \infty} \frac{8x+20}{18x-6} \ (masih\ \frac{\infty}{\infty} )[/tex]
[tex]= \lim_{x \to \infty} \frac{8}{18}[/tex]
[tex]=\frac{8}{18}[/tex]
[tex]=\frac{4}{9}[/tex]
8.
[tex]= \lim_{x \to \infty} (\sqrt{x^{2} -6x+5} -\sqrt{x^{2} +x-1} )\times\frac{ (\sqrt{x^{2} -6x+5} +\sqrt{x^{2} +x-1})}{ (\sqrt{x^{2} -6x+5} +\sqrt{x^{2} +x-1})}[/tex]
[tex]= \lim_{x \to \infty} \frac{ x^{2} -6x+5 -(x^{2} +x-1)}{ (\sqrt{x^{2} -6x+5} +\sqrt{x^{2} +x-1})}[/tex]
[tex]= \lim_{x \to \infty} \frac{ -7x+6 }{ (\sqrt{x^{2} -6x+5} +\sqrt{x^{2} +x-1})}\times\frac{\frac{1}{x} }{\frac{1}{x} }[/tex]
[tex]= \lim_{x \to \infty} \frac{ -7+\frac{6}{x} }{ \sqrt{\frac{x^{2} }{x^{2} } -\frac{6x}{x^{2} } +\frac{5}{x^{2} } } +\sqrt{\frac{x^{2} }{x^{2} } +\frac{x}{x^{2} } -\frac{1}{x^{2} } }}[/tex]
[tex]=\frac{ -7+0 }{ \sqrt{1 -0+0 } +\sqrt{1+0-0}}[/tex]
[tex]=-\frac{7}{2}[/tex]
9.
[tex]= \lim_{x \to \infty} (\sqrt{2x^{2} -4x+1} -\sqrt{2x^{2} +2x-1} )\times\frac{ (\sqrt{2x^{2} -4x+1} +\sqrt{2x^{2} +2x-1})}{ \sqrt{2x^{2} -4x+1} +\sqrt{2x^{2} +2x-1} )}[/tex]
[tex]= \lim_{x \to \infty} \frac{ 2x^{2} -4x+1 -(2x^{2} +2x-1)}{ (\sqrt{2x^{2} -4x+1} +\sqrt{2x^{2} +2x-1})}[/tex]
[tex]= \lim_{x \to \infty} \frac{ -6x+2 }{ (\sqrt{2x^{2} -4x+1} +\sqrt{2x^{2} +2x-1})}\times\frac{\frac{1}{x} }{\frac{1}{x} }[/tex]
[tex]= \lim_{x \to \infty} \frac{ -6+\frac{6}{x} }{ \sqrt{\frac{2x^{2} }{x^{2} } -\frac{4x}{x^{2} } +\frac{1}{x^{2} } } +\sqrt{\frac{2x^{2} }{x^{2} } +\frac{2x}{x^{2} } -\frac{1}{x^{2} } }}[/tex]
[tex]=\frac{ -6+0 }{ \sqrt{2 -0+0 } +\sqrt{2+0-0}}[/tex]
[tex]=-\frac{3}{\sqrt{2} }[/tex]
[tex]=-\frac{3}{2} \sqrt{2}[/tex]
10.
[tex]= \lim_{x \to \infty} (\sqrt{x^{2} +5x-1} -\sqrt{x^{2} } )\times\frac{ (\sqrt{x^{2} +5x-1} +\sqrt{x^{2} })}{ (\sqrt{x^{2} +5x-1} +\sqrt{x^{2} })}[/tex]
[tex]= \lim_{x \to \infty} \frac{ x^{2} +5x-1 -x^{2} }{ (\sqrt{x^{2} +5x-1} +\sqrt{x^{2} })}[/tex]
[tex]= \lim_{x \to \infty} \frac{ 5x-1 }{(\sqrt{x^{2} +5x-1} +\sqrt{x^{2} })}\times\frac{\frac{1}{x} }{\frac{1}{x} }[/tex]
[tex]= \lim_{x \to \infty} \frac{ 5-\frac{1}{x} }{ \sqrt{\frac{x^{2} }{x^{2} } +\frac{5x}{x^{2} } -\frac{1}{x^{2} } } +\sqrt{\frac{x^{2} }{x^{2} } } }}[/tex]
[tex]=\frac{ 5-0 }{ \sqrt{1 +0-0 } +\sqrt{1}}[/tex]
[tex]=\frac{5}{2}[/tex]
11.
[tex]= \lim_{x \to \infty} \sqrt{9x^{2} -2} -(3x+1)[/tex]
[tex]= \lim_{x \to \infty} (\sqrt{9x^{2} -2} -(3x+1) } )\times\frac{ \sqrt{9x^{2} -2} +(3x+1)}{ \sqrt{9x^{2} -2} +(3x+1)}[/tex]
[tex]= \lim_{x \to \infty} \frac{ 9x^{2} -2 -(9x^{2}+6x+1) }{ \sqrt{9x^{2} -2} +(3x+1)}[/tex]
[tex]= \lim_{x \to \infty} \frac{ -6x-3 }{ \sqrt{9x^{2} -2} +(3x+1)}\times\frac{\frac{1}{x} }{\frac{1}{x} }[/tex]
[tex]= \lim_{x \to \infty} \frac{ -6-\frac{3}{x} }{ \sqrt{\frac{9x^{2} }{x^{2} } -\frac{2}{x^{2} } } +\frac{3x }{x } +\frac{1}{x} }}[/tex]
[tex]=\frac{ -6-0 }{ \sqrt{9-0 } +3+0}[/tex]
[tex]=-1[/tex]
12 & 13 kurang yakin
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