Evaluate the limit x tends to zerolog(1+5x) whole divided by x.

$lim _{xto :0}left(frac{ln left(1+5xright)}{x}right)$

Apply Lhospital rule,

$=lim _{xto :0}left(frac{frac{5}{1+5x}}{1}right)$ $=lim _{xto :0}left(frac{5}{1+5x}right)$

Plug in the value x=0

$=frac{5}{1+5cdot :0}$ =5