Question

Найти пределы функций

Answer 1

$1); limlimits _{x to infty}frac{sqrt[3]{2x^6+1}-x^2+1}{sqrt{x^4+2}+sqrt{x^3+1}}=Big [frac{:x^2}{:x^2}Big ]=limlimits _{x to infty}frac{sqrt[3]{2+frac{1}{x^6}}-1+frac{1}{x^2}}{sqrt{1+frac{2}{x^2}}+sqrt{frac{1}{x}+frac{1}{x^4}}}=\\=frac{sqrt[3]2-1+0}{1+0}=sqrt[3]2-1$

$2); limlimits _{x to 3}frac{x^2-5x+6}{x^2-8x+15}=limlimits _{x to 3}frac{(x-3)(x-2)}{(x-3)(x-5)}=limlimits _{x to infty}frac{x-2}{x-5}=frac{3-2}{3-5}=-0,5\\3); limlimits _{x to 0}frac{2, sinx+arctgx}{tgx-2, arcsinx}=limlimits _{x to 0}frac{2, cosx+frac{1}{1+x^2}}{frac{1}{cos^2x}-frac{2}{sqrt{1-x^2}}}=frac{2+1}{1-2}=-3$

$4); star ; limlimits _{x to 0}Big (frac{1+x}{2+x}Big )^{frac{1-sqrt{x}}{1-x}}=(frac{1}{2})^{frac{1-0}{1+0}}=(frac{1}{2})^1=frac{1}{2}\\star ; star ; limlimits _{x to infty}Big (frac{1+x}{2+x}Big )^{frac{1-sqrt{x}}{1-x}}=[, 1^{infty , }]=limlimits _{x to infty}Big ((1+frac{-1}{x+2})^{frac{x+2}{-1}}Big )^{frac{-1}{x+2}cdot frac{1-sqrt{x}}{(1-sqrt{x})(1+sqrt{x})}}=\\=e^{limlimits _{x to infty}frac{-1}{(x+2)(1+sqrt{x})}}=Big [e^{frac{-1}{infty }}Big ]=e^0=1$