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SMUP 2010

Solusi SMUP 2010 Kode 041 – Nomor 24


Nilai\ dari
\lim \limits_{x \rightarrow -\infty} \frac{\sqrt{x^2+x}}{2x-1}
adalah\ ....
A.\quad  1\\B.\quad -1\\C.\quad 0\\D.\quad \frac{1}{2}\\E.\quad -\frac{1}{2}
Jawaban: D

\begin{array}{rcl}\lim \limits_{x \rightarrow -\infty} \frac{\sqrt{x^2+x}}{2x-1}&=&\lim \limits_{x \rightarrow -\infty} \frac{\sqrt{x^2+x}}{\sqrt{(2x-1)^2}}\\\\&=&\lim \limits_{x \rightarrow -\infty} \sqrt{\frac{x^2+x}{4x^2-4x+1}}\\\\&=&\sqrt{\lim \limits_{x \rightarrow -\infty} \frac{x^2+x}{4x^2-4x+1}}\\\\&=&\sqrt{\lim \limits_{x \rightarrow -\infty} \frac{1+\frac{1}{x}}{4-\frac{4}{x}+\frac{1}{x^2}}}\\\\&=&\sqrt{\frac{1-0}{4+0+0}}\\\\&=&\sqrt{\frac{1}{4}}\\\\&=&\frac{1}{2}\end{array}

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