Question

Найти пределы функций

Answer 1

$1); ; limlimits _{x to 3}frac{2x^2-5x-3}{3x^2-4x-15}=limlimits _{x to 3}frac{2(x+frac{1}{2})(x-3)}{3(x+frac{5}{3})(x-3)}=limlimits _{x to 3}frac{2x+1}{3x+5}=frac{2cdot 3+1}{3cdot 3+5}=frac{7}{14} =0,5\\\limlimits _{x to infty }frac{2x^2-5x-3}{3x^2-4x-15}=limlimits _{x to infty}frac{2-frac{5}{x}-frac{3}{x^2}}{3-frac{4}{x}-frac{15}{x^2}}=frac{2}{3}\\\2); ; limlimits _{x to 0}frac{sqrt{x+4}-2}{x}=limlimits _{x to 0}frac{(x+4)-4}{x, (sqrt{x+4}+2)}=limlimits _{x to 0}frac{1}{sqrt{x+4}+2}=frac{1}{2+2}=0,25$

$3); ; limlimits _{x to 0}frac{1-cos^24x}{4x}=limlimits _{x to 0}frac{sin^24x}{4x}=[; sinalpha sim alpha ; ,; esli; ; alpha to 0; ]=\\=limlimits_{x to 0}frac{(4x)^2}{4x}=limlimits_{x to 0}(4x)=0\\\4); ; limlimits _{x to infty}Big (frac{x+9}{x-3}Big )^{x}=Big [, 1^{infty }, Big ]=limlimits_{x to infty}Big (Big (1+frac{12}{x-3}Big )^{frac{x-3}{12}}Big )^{frac{12, x}{x-3}}=\\\=e^{limlimits _{x to infty}frac{12, x}{x-3}}=e^{12}$