a) [latex]4xge0 wedge x-5>0 \ xge0 wedge x>5 \ D: x>5[/latex] [latex] lim_{x o 5^+} f(x)= lim_{x o 5^+}frac{sqrt{4x}}{sqrt{x-5}}=frac{sqrt{20}}{sqrt{0^+}}}=+infty[/latex] zatem prosta [latex]x=5[/latex] jest asymptotą pionową jednostronną [latex] lim_{x o infty} f(x)= lim_{x o infty} frac{sqrt{4x}}{sqrt{x-5}}= lim_{x o infty}frac{sqrt{x}cdot 2}{sqrt{x}cdot (sqrt{1-frac5{x}})}= \ = lim_{x o infty}frac2{sqrt{1-frac5x}}=frac2{sqrt{1-0}}=2[/latex] [latex]y=2[/latex] to asymptota pozioma prawostronna b) [latex]x^2+2x-3 e0 \ Delta=4-4cdot 1cdot (-3)=4+12=16 o sqrt{Delta}=4 \ x_1=frac{-2-4}2=-3 , x_2=frac{-2+4}2=1 \ D: xin Rsetminus {-3;1}[/latex] asymptoty liczymy granice dla x dążącego do -3 oraz do 1 [latex] lim_{n o -3^-} frac{2x+7}{(x-1)(x+3)}=frac1{-4cdot (0^-)}=frac1{0^+}=+infty \ lim_{n o -3^+} frac{2x+7}{(x-1)(x+3)}=frac1{-4cdot (0^+)}=frac1{0^-}=-infty[/latex] [latex]x=-3[/latex] - asymptota pionowa obustronna [latex] lim_{x o 1^-} frac{2x+7}{(x-1)(x+3)}=frac{9}{(0^-)cdot 4}=frac9{0^-}=-infty \ lim_{x o 1^+} frac{2x+7}{(x-1)(x+3)}=frac9{(0^+)cdot 4}=frac9{0^+}=+infty[/latex] [latex]x=1[/latex] - asymptota pionowa obustronna szukamy asymptot poziomych - liczymy granice w - i w +nieskończoności [latex] lim_{x o infty} frac{2x+7}{x^2+2x-3}= lim_{x o infty} frac{x(2+frac7{x})}{x^2(1+frac2{x}-frac3{x^2})}= lim_{x o infty} frac{2+frac7{x}}{x(1+frac2{x}-frac3{x^2})}=\=frac{2+0}{(+infty)(1+0-0)}=frac2{+infty}=0 \ lim_{x o -infty} frac{2x+7}{x^2+2x-3}= lim_{x o -infty} frac{x(2+frac7{x})}{x^2(1+frac2{x}-frac3{x^2})}= lim_{x o -infty} frac{2+frac7{x}}{x(1+frac2{x}-frac3{x^2})}=\=frac{2+0}{(-infty)(1+0-0)}=frac2{-infty}=0[/latex] [latex]y=0[/latex] to asymptota pozioma obustronna
