Evaluate lim(x→0)[log(6+x)-log(6-x)]/x

Question

Evaluate :

\(\lim\limits_{x \to 0}\frac{log(6+x)-log(6-x)}{x}\)

Answer 1

\(\lim\limits_{x \to 0}\frac{log(1+\frac{x}{6})-log6(1-\frac{x}{6})}{x}\)

= \(\lim\limits_{x \to 0}\frac{[log6+log(1+\frac{x}{6}]-[log6+log(1-\frac{x}{6})]}{x}\)

= \(\lim\limits_{x \to 0}[\frac{log(1+\frac{x}{6})}{x}-\frac{log(1-\frac{x}{6})}{x}]\)

= \(\lim\limits_{x \to 0}.\frac{1}{6}\frac{log(1+\frac{x}{6})}{\frac{x}{6}}+\lim\limits_{x \to 0}.\frac{1}{6}\frac{log(1-\frac{x}{6})}{(-\frac{x}{6})}]\)

= \(\frac{1}{6}\times1+\frac{1}{6}\times1\,[∵ \lim\limits_{x \to 0}\frac{log(1+x)}{x}=\lim\limits_{x \to 0}\frac{log(1-x)}{-x}=1]\)

= 0