Question

Помогите решить пределы пожалуйста.

Answer 1

$lim_{x to infty} ( frac{x^2+4}{x^2+5})^{-3x^2}={1^{-infty}}= lim_{x to infty} ( frac{x^2+5-1}{x^2+5} )^{-3x^2}=\ = lim_{x to infty} (1+ frac{-1}{x^2+5} )^{-3x^2}= lim_{x to infty} (1+ frac{-1}{x^2+5} )^{-3x^2cdot frac{-1}{x^2+5}cdot frac{x^2+5}{-1} }=\ = e^{ lim_{x to infty} frac{3x^2}{x^2+5}}=e^3$

$lim_{x to 1} frac{ln (1+x)-ln 2}{x-1} ={ frac{0}{0}}= lim_{x to 1} frac{ln frac{1+x}{2} }{x-1} = lim_{x to 1} frac{ln frac{2-1+x}{2} }{x-1} =\ = lim_{x to 1} frac{ln (1+ frac{x-1}{2}) }{x-1} = lim_{x to 1} frac{ln(1+ frac{x-1}{2}) }{2cdot frac{x-1}{2} } = frac{1}{2}$