Minggu, 14 Oktober 2012

Limit Aljabar

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SPMB 2003 regional III
\displaystyle \lim_{x \to \infty} \left( 3x - 2 - \sqrt{9x^2 - 2x + 5} \right) = ...
(A)   -\frac{5}{6}
(B)   -2\frac{1}{3}
(C)   -1\frac{2}{3}
(D)   2\frac{1}{3}
(E)   \frac{5}{6}

Jawab :
\begin{aligned} \displaystyle \lim_{x \to \infty} \left( 3x - 2 - \sqrt{9x^2 - 2x + 5} \right) &= \lim_{x \to \infty} \left( 3x - \sqrt{9x^2 - 2x + 5} \right) - 2 \\ &= \lim_{x \to \infty} \left( \sqrt{(3x)^2} - \sqrt{9x^2 - 2x + 5} \right) - 2 \\ &= \frac{0 - (-2)}{2\sqrt{9}} - 2 \\ &= -1\frac{2}{3} \end{aligned}
\therefore   Jawab : C
catatan :
\displaystyle \boxed{ \lim_{x \to \infty} \left( \sqrt{ax^2+bx+c}-\sqrt{ax^2+px+q} \right) = \frac{b-p}{2\sqrt{a}}}

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