Question

Решите предел,пожалуйста

Answer 1

Пример на второй замечательный предел.

$lim_{x to inft{-4}} (9+2x)^{ frac{x+1}{4x+x^2} } = lim_{x to inft{-4}} (1+(8+2x))^{ frac{x+1}{x(4+x)} } = \ \ = lim_{x to inft{-4}} [(1+(8+2x))^{ frac{1}{8+2x}*(8+2x) } ]^{ frac{x+1}{x(4+x)} } = \ \ = lim_{x to inft{-4}} [(1+(8+2x))^{ frac{1}{8+2x} } ]^{ frac{2(4+x)(x+1)}{x(4+x)} } = \ \ = lim_{x to inft{-4}} [(1+(8+2x))^{ frac{1}{8+2x} } ]^{ frac{2(x+1)}{x} } =$

$= lim_{x to inft{-4}} [(1+(8+2x))^{ frac{1}{8+2x} } ]^{lim_{x to inft{-4}} frac{2(x+1)}{x} } = \ \ e^{lim_{x to inft{-4}} frac{2(x+1)}{x} } = e^{ frac{2(-4+1)}{-4} } = e^{ frac{3}{2} }$