Question

Помогите с предалами,даю 50 баллов

Answer 1

$1)quad limlimits _{x to 0} frac{x+2}{sinx} =frac{0+2}{sin0}=left [; frac{2}{0}; right ]=infty \\2)quad limlimits _{x to 2} frac{x^2-3x+2}{sqrt{5-x}-sqrt{x+1}} = limlimits_{x to 2} frac{(x-1)(x-2)(sqrt{5-x}+sqrt{x+1})}{(sqrt{5-x}-sqrt{x+1})(sqrt{5-x}+sqrt{x+1})} =\\= limlimits _{x to 2} frac{(x-1)(x-2)(sqrt{5-x})(sqrt{x+1})}{5-x-(x+1)} =limlimits _{x to 2} frac{(x-1)(x-2)(sqrt{5-x}+sqrt{x+1})}{-2(x-2)}=$
 
 $=limlimits _{xto 2}frac{(x-1)(sqrt{5-x}+sqrt{x+1})}{-2}=frac{sqrt3+sqrt3}{-2}=-frac{2sqrt3}{2}=-sqrt3$

$3)quad limlimits _{x to infty} (sqrt{x^3+3}-2sqrt{x^2+1})=\\=limlimits _{xto infty } frac{(sqrt{x^3+3}-2sqrt{x^2+1})(sqrt{x^3+3}+2sqrt{x^2+1})}{sqrt{x^3+3}+2sqrt{x^2+1}} =limlimits _{xto infty }frac{x^3+3-4(x^2+1)}{sqrt{x^2+3}+2sqrt{x^2+1}}=\\=limlimits _{xto infty }frac{x^3}{x^{3/2}}=limlimits _{xto infty }x^{3/2}=infty$

$4)quad limlimits _{x to infty} left (frac{x+3-1+1}{x+2}right )^{x}= limlimits _{x to infty} left (frac{(x+2)+1}{x+2}right )^{x}=\\=limlimits _{xto infty }left (left (1+frac{1}{x+2}right )^{x+2}right )^{frac{x}{x+2}}=e^{limlimits _{xto infty }frac{x}{x+2}}=e^1=e$

$5)quad limlimits _{x to 4}(5-x)^{frac{3x}{x-4}}=limlimits _{xto 4}(1+(4-x))^{frac{3x}{x-4}} =\\=limlimits _{xto 4}left (left (1+underbrace {(4-x)}_{to 0 }right )^{frac{1}{4-x}}right )^{frac{(4-x)3x}{-(4-x)}}=e^{limlimits _{xto 4}(-3x)}=e^{-12}$