limx→11+logx-x1-2x+x2 equals

$\lim_{x \to 1} \frac{1 + \log x - x}{1 - 2 x + x^{2}}$ equals

A
1
B
0
C
-1
D
– 1/2

Solution:

$\lim_{x \to 1} \frac{1 + \log x - x}{1 - 2 x + x^{2}} = \lim_{x \to 1} \frac{1 / x - 1}{- 2 + 2 x}$

(using L ’Hôpital Rule)

$= \lim_{x \to 1} - \frac{1}{2} \frac{(1 - x)}{x (1 - x)} = - \frac{1}{2} \lim_{x \to 1} \frac{1}{x} = - \frac{1}{2}$

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