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Find the limit as x goes to 1 of log x / (x2 + x – 2)

by RoRi

Evaluate the limit.

    \[ \lim_{x \to 1} \frac{\log x}{x^2 + x -2}. \]

From this exercise (Section 7.11, Exercise #4) we know

    \[ \log x = (x-1) - \frac{1}{2}(x-1)^2 + o((x-1)^2). \]

Therefore,

    \begin{align*} \lim_{x \to 1} \frac{\log x}{x^2+x-2} &= \lim_{x \to 1} \frac{(x-1) - \frac{1}{2}(x-1)^2 + o((x-1)^2)}{x^2+x-2} \\[9pt] &= \lim_{x \to 1} \left( \frac{1}{x+2} + \frac{(x-1)}{2(x+2)} + \frac{o(x-1)^2}{x-1}\cdot \frac{1}{x+2} \right) \\[9pt] &= \frac{1}{3} + 0 + 0 \cdot \frac{1}{3} \\[9pt] &= \frac{1}{3}. \end{align*}

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