Question

Find the lim [(ln(x+5) - ln5) / x ] as x approaches 0 ...?

Answer 1

Use $\lim_{x \to 0}\frac{ \ln{( 1+x )}}{x}=1$

$\lim_{x \to 0}\frac{\ln(x+5) - \ln5}{x} \\ \\ \lim_{x \to 0}\frac{ \ln{ \frac{x+5}{5} }}{x} \\ \\ \lim_{x \to 0}\frac{ \ln{( 1+\frac{x}{5} )}}{x} \\ \\ \lim_{x \to 0}\frac{ \ln{( 1+\frac{x}{5} )}}{\frac{x}{5}\times 5} \\ \\ \frac{1}{5} \lim_{x \to 0}\frac{ \ln{( 1+\frac{x}{5} )}}{\frac{x}{5}} \\ \\ \frac{1}{5} \times 1 \\ \\\frac{1}{5}$