Question

пожалуйста решите примеры на пределы

Answer 1

7. $lim_{x to 1} frac{ sqrt{x}-1 }{x-1} =lim_{x to 1} frac{ sqrt{x}-1 }{(sqrt{x}-1)(sqrt{x}+1)}= lim_{x to 1} frac{ 1 }{sqrt{x}+1}= frac{1}{2}$
8. $lim_{x to 5} frac{ sqrt{x+1}- sqrt{6} }{ sqrt{x-1}-2 } =lim_{x to 5} frac{( sqrt{x+1}- sqrt{6})(sqrt{x-1}+2) }{ (x-1)-4 }=$
$=lim_{x to 5} frac{ sqrt{x^2-1} +2sqrt{x+1}-sqrt{6}sqrt{x-1}-2sqrt{6} }{ x-5 }=lim_{x to 5} frac{ 2 +2sqrt{6}-2sqrt{6}-2sqrt{6} }{ x-5 }=$
=$lim_{x to 5} frac{ 2 +2sqrt{6}-2sqrt{6}-2sqrt{6} }{ x-5 }=lim_{x to 5} frac{ 2 -2sqrt{6} }{ x-5 }= infty$
9. $lim_{xto 0} frac{cos8x-1}{xtg5x}=lim_{xto 0} frac{1-(8x)^2/2-1}{x*5x}= lim_{xto 0} frac{-32x^2}{5x^2}=-32/5=-6 frac{2}{5}$
10. $lim_{x to 0} frac{sin(x/15)}{x} = lim_{x to 0} frac{x/15}{x}=1/15$
11. $lim_{x to 0} frac{tg2x}{x}= lim_{x to 0} frac{2x}{x}=2$
12. $lim_{x to infty} (frac{2x-1}{2x+4} )^{-x}=lim_{x to infty} (frac{2x+4}{2x-1} )^x=lim_{x to infty} (frac{2x-1+5}{2x-1} )^x=$
$=lim_{x to infty} (1+frac{5}{2x-1} )^x==lim_{x to infty} (1+frac{5}{2x-1} )^{frac{2x-1}{5}{frac{5}{2x-1} x}}=e^{5/2}$