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Rationalization Method to Remove Indeterminate Form

limx→ -2x3-7x...

Question

$lim x \to - 2 \frac{x^{3} - 7 x - 6}{x^{4} + 5 x - 6}$

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Solution

$y = lim x \to - 2 \frac{x^{3} - 7 x - 6}{x^{+} 5 x - 6}$
using L'Hospital rule we have
$y = lim x \to - 2 \frac{\frac{d}{d x} (x^{3} - 7 x - 6)}{\frac{d}{d x} (x^{4} + 5 x - 6)} = \frac{3 x^{2} - 7}{4 x^{3} + 5}$
Now put limit
$y = \frac{3 (- 2)^{2} - 7}{4 (- 2)^{3} + 5} = \frac{12 - 7}{- 32 + 5}$
$[y = \frac{- 5}{27}]$ .

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