Oblicz granice: [latex] lim_{n o 6} frac{-x^{2} +10x-24 }{ x^{2} -2x-24} [/latex] [latex] lim_{n o -2} frac {x^{3} +8 }{ x^{2} -4} [/latex] [latex] lim_{n o 5} frac{125- x^{3} }{4 x^{2} -100} [/latex] [latex] lim_{n o -3} frac{ x^{4}-8 x^{2} -9 }

Nie przepisuję przykładów. [latex]=limlimits_{x o6}dfrac{-(x^2-10x+24)}{x^2-6x+4x-24}=limlimits_{x o6}dfrac{-(x^2-6x-4x+24)}{x(x-6)+4(x-6)}\\=limlimits_{x o6}dfrac{-[x(x-6)-4(x-6)]}{(x-6)(x+4)}=limlimits_{x o6}dfrac{-(x-6)(x-4)}{(x-6)(x+4)}\\=limlimits_{x o6}dfrac{-(x-4)}{x+4}=dfrac{-(6-4)}{6+4}=dfrac{-2}{10}=-dfrac{1}{5}[/latex] [latex]=limlimits_{x o-2}dfrac{x^3+2^3}{x^2-2^2}=limlimits_{x o-2}dfrac{(x+2)(x^2-2x+2^2)}{(x-2)(x+2)}=limlimits_{x o-2}dfrac{x^2-2x+4}{x-2}\\=dfrac{(-2)^2-2cdot(-2)+4}{-2-2}=dfrac{4+4+4}{-4}=dfrac{12}{-4}=-3 [/latex] [latex]=limlimits_{x o5}dfrac{125-x^3}{4x^2-100}=limlimits_{x o5}dfrac{-(x^3-125)}{4(x^2-25)}=limlimits_{x o5}dfrac{-(x^3-5^3)}{4(x^2-5^2)}\\=limlimits_{x o5}dfrac{-(x-5)(x^2+5x+5^2)}{4(x-5)(x+5)}=limlimits_{x o5}dfrac{-(x^2+5x+25)}{4(x+5)}\\=dfrac{-(5^2+5cdot5+25)}{4(5+5)}=dfrac{-(25+25+25)}{4cdot10}=-dfrac{75}{40}=-dfrac{15}{8}=-1dfrac{7}{8}[/latex] [latex]=limlimits_{x o-3}dfrac{x^4-8x^2-9}{x^3+3x^2}=limlimits_{x o-3}dfrac{x^4-9x^2+x^2-9}{x^2(x+3)}\\=limlimits_{x o-3}dfrac{x^2(x^2-9)+1(x^2-9)}{x^2(x+3)}=limlimits_{x o-3}dfrac{(x^2-9)(x^2+1)}{x^2(x+3)}\\=limlimits_{x o-3}dfrac{(x^2-3^2)(x^2+1)}{x^2(x+3)}=limlimits_{x o-3}dfrac{(x-3)(x+3)(x^2+1)}{x^2(x+3)}\\=limlimits_{x o-3}dfrac{(x-3)(x^2+1)}{x^2}=dfrac{(-3-3)[(-3)^2+1]}{(-3)^2}\\=dfrac{(-6)(9+1)}{9}=dfrac{-2cdot10}{3}=-dfrac{20}{3}=-6dfrac{2}{3}[/latex] [latex]limlimits_{x o-1}dfrac{x^2-5x^2+4}{x^4+2x^2-8}=dfrac{(-1)^2-5cdot(-1)^2+4}{(-1)^4+2cdot(-1)^2-8}=dfrac{1-5+4}{1+2-8}=dfrac{0}{-5}=0[/latex] W ostatnim mogliśmy podstawić liczbę ponieważ w mianowniku nie otrzymujemy 0.