lim-x-2-log-x-2-1-log-2-1-x-1-

Question Number 66478 by hmamarques1994@gmail.com last updated on 15/Aug/19

\underset{x \to 2}{l i m} [\frac{{l o g}_{x} (2) - 1}{{l o g}_{2} (\frac{1}{x}) + 1}] = ?

Commented by gunawan last updated on 16/Aug/19

\lim_{x \to 2} \frac{\log_{x} 2 - \log_{x} x}{\log_{2} (\frac{1}{x}) + \log_{2} 2}

= \lim_{x \to 2} \frac{\log_{x} (\frac{2}{x})}{\log_{2} (\frac{1}{2 x})} = \frac{\log_{2} (\frac{2}{2})}{\log_{2} (\frac{1}{4})} = \frac{\log_{2} 1}{- 2 \times \log_{2}} = 0

Commented by Rasheed.Sindhi last updated on 16/Aug/19

\log_{2} (\frac{1}{x}) + \log_{2} 2 = \log_{2} (\frac{1}{x} \times 2) = \log_{2} (\frac{2}{x})