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L'Hospital Rule to Remove Indeterminate Form

ltx→ 2 1x-2 -...

Question

$l t_{x \to 2} (\frac{1}{x - 2} - \frac{4}{x^{2} - 4})$

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Solution

$\frac{1}{x - 2} - \frac{4}{x^{2} - 4} = \frac{1}{x - 2} - \frac{4}{(x - 2) (x + 2)}$
$= \frac{x + 2 - 4}{(x - 2) (x + 2)} = \frac{x - 2}{(x - 2) (x + 2)} = \frac{1}{x + 2}$
$lim x \to 2 (\frac{1}{x - 2} - \frac{4}{x^{2} - 4}) = lim x \to 2 \frac{1}{x + 2} = \frac{1}{2 + 2} = \frac{1}{4}$

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