Question

Решить предел :
lim = ((3*x+5)(ln(x+5)-ln(x))
x->+oo

Answer 1

$lim_{n to infty} (3x+5)(ln(x+5)-lnx) = lim_{n to infty} (3x+5)ln frac{x+5}{x} = \ \ = lim_{n to infty} ln (frac{x+5}{x} )^{3x+5}= lim_{n to infty} ln (1+frac{5}{x} )^{3x+5}= \ \ = ln (lim_{n to infty} (1+frac{5}{x} )^{3x+5}) = ln (lim_{n to infty} (1+frac{5}{x} )^{ frac{x}{5} frac{5}{x} (3x+5)}) = \ \ = ln ((lim_{n to infty} (1+frac{5}{x} )^ frac{x}{5} )^{ 15+ frac{5}{x} }) =$

$= ln ((lim_{n to infty} (1+frac{5}{x} )^ frac{x}{5} )^{ lim_{n to infty}(15+ frac{5}{x}) }) = ln e^{ lim_{n to infty}(15+ frac{5}{x}) } = \ \ = lim_{n to infty}(15+ frac{5}{x}) = 15$